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Class-XII-Maths Continuity and Differentiabil ity
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CBSE NCERT Solutions for Class 12 Maths Chapter 05
Back of Chapter Questions
Exercise π. π
1. Prove that the function π(π₯) = 5π₯ β 3 is continuous at π₯ = 0, at π₯ = β3 and at π₯ = 5.
Solution:
Given function is π(π₯) = 5π₯ β 3
At π₯ = 0, π(0) = 5(0) β 3 = β3
LHL = limπ₯β0β
π(π₯) = limπ₯β0β
(5π₯ β 3) = β3
RHL = limπ₯β0+
π(π₯) = limπ₯β0+
(5π₯ β 3) = β3
Here, at π₯ = 0, LHL = RHL = π(0) = β3
Hence, the function π is continuous at π₯ = 0.
Now at π₯ = β3, π(β3) = 5(β3) β 3 = β18
LHL = limπ₯ββ3β
π(π₯) = limπ₯ββ3β
(5π₯ β 3) = β18
RHL = limπ₯ββ3+
π(π₯) = limπ₯ββ3+
(5π₯ β 3) = β18
Here, at π₯ = β3, LHL = RHL = π(β3) = β18
Hence, the function π is continuous at π₯ = β3.
At π₯ = 5, π(5) = 5(5) β 3 = 22
πΏπ»πΏ = limπ₯β5β
π(π₯) = limπ₯β5β
(5π₯ β 3) = 22
RHL = limπ₯β5+
π(π₯) = limπ₯β5+
(5π₯ β 3) = 22
Here, at π₯ = 5, LHL = RHL = π(5) = 22
Hence, the function π is continuous at π₯ = 5.
2. Examine the continuity of the function π(π₯) = 2π₯2 β 1 at π₯ = 3.
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Solution:
Given function is π(π₯) = 2π₯2 β 1.
At π₯ = 3, π(3) = 2(3)2 β 1 = 17
LHL = limπ₯β3β
π(π₯) = limπ₯β3β
(2π₯2 β 1) = 17
RHL = limπ₯β3+
π(π₯) = limπ₯β3+
(2π₯2 β 1) = 17
Here, at π₯ = 3, LHL = RHL = π(3) = 17
Therefore, the function π is continuous at π₯ = 3.
3. Examine the following functions for continuity:
(a) π(π₯) = π₯ β 5
(b) π(π₯) =1
π₯β5, π₯ β 5
(c) π(π₯) =π₯2β25
π₯+5, π₯ β β5
(d) π(π₯) = |π₯ β 5|
Solution:
(a) Given function π(π₯) = π₯ β 5
Let π be any real number. At π₯ = π, π(π) = π β 5
LHL = limπ₯βπβ
π(π₯) = limπ₯βπβ
(π₯ β 5) = π β 5
RHL = limπ₯βπ+
π(π₯) = limπ₯βπ+
(π₯ β 5) = π β 5
At π₯ = π, LHL = RHL = π(π) = π β 5
Hence, the function π is continuous for all real numbers.
(b) Given function π(π₯) =1
π₯β5, π₯ β 5
Let π(π β 5) be any real number. At π₯ = π, π(π) =1
πβ5
LHL = limπ₯βπβ
π(π₯) = limπ₯βπβ
(1
π₯β5) =
1
πβ5
RHL = limπ₯βπ+
π(π₯) = limπ₯βπ+
(1
π₯β5) =
1
πβ5
At π₯ = π, LHL = RHL = π(π) =1
πβ5
Hence, the function π is continuous for all real numbers (except 5).
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(c) Given function π(π₯) =π₯2β25
π₯+5, π₯ β β5
Let π(π β β5) be any real number.
At π₯ = π, π(π) =π2β25
π+5=
(π+5)(πβ5)
(π+5)= (π + 5)
LHL = limπ₯βπβ
π(π₯) = limπ₯βπβ
(π₯2β25
π₯+5) = lim
π₯βπβ((π+5)(πβ5)
(π+5)) = π + 5
RHL = limπ₯βπ+
π(π₯) = limπ₯βπ+
(π₯2β25
π₯+5) = lim
π₯βπ+((π+5)(πβ5)
(π+5)) = π + 5
At π₯ = π, LHL = RHL = π(π) = π + 5
Hence, the function π is continuous for all real numbers (except β5).
(d) Given function is π(π₯) = |π₯ β 5| = {5 β π₯, π₯ < 5π₯ β 5, π₯ β₯ 5
Let π be any real number. According to question,π can be π < 5 or π = 5 or π > 5.
First case: If π < 5,
π(π) = 5 β π and limπ₯βπ
π(π₯) = limπ₯βπ
(5 β π₯) =5 β π. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 5.
Second case: If π = 5,
π(π) = π β 5 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯ β 5) = π β 5. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous at π₯ = 5.
Third case: If π > 5,
π(π) = π β 5 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯ β 5) = π β 5. Here, limπ₯βπ
π(π₯) =π(π)
Hence, the function π is continuous for all real numbers greater than 5.
Hence, the function π is continuous for all real numbers.
4. Prove that the function π(π₯) = π₯π, is continuous at π₯ = π, where π is a positive integer.
Solution:
Given function is π(π₯) = π₯π.
At π₯ = π, π(π) = π₯π
limπ₯βπ
π(π₯) = limπ₯βπ
(π₯π) = π₯π
Here, at π₯ = π, limπ₯βπ
π(π₯) =π(π) = π₯π
Since limπ₯βπ
π(π₯) =π(π) = π₯π
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Hence, the function π is continuous at π₯ = π, where π is positive integer.
5. Is the function π defined by π(π₯) = {π₯, π₯ β€ 15, π₯ > 1
continuous at π₯ = 0? At π₯ = 1? At π₯ = 2?
Solution:
Given function is π(π₯) = {π₯, π₯ β€ 15, π₯ > 1
At π₯ = 0, π(0) = 0
limπ₯β0
π(π₯) = limπ₯β0
(π₯) = 0
Here at π₯ = 0, limπ₯β0
π(π₯) = π(0) = 0
Hence, the function π is discontinuous at π₯ = 0.
At π₯ = 1, π(1) = 1
LHL = limπ₯β1β
π(π₯) = limπ₯β1β
(π₯) = 1
RHL = limπ₯β1+
π(π₯) = limπ₯β1+
(5) =5
Here, at π₯ = 1, LHL β RHL.
Hence, the function π is discontinuous at π₯ = 1.
At π₯ = 2, π(2) = 5
limπ₯β2
π(π₯) = limπ₯β2
(5) = 5
Here, at π₯ = 2, limπ₯β2
π(π₯) = π(2) = 5
Hence, the function π is continuous at π₯ = 2.
6. Find all points of discontinuity of π, where π is defined by π(π₯) = {2π₯ + 3, If π₯ β€ 22π₯ β 3, If π₯ > 2
Solution:
Let π be any real number. According to question, π < 2 or π = 2 or π > 2
First case: π < 2
π(π) = 2π + 3 and limπ₯βπ
π(π₯) = limπ₯βπ
(2π₯ + 3) = 2π + 3. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers smaller than 2.
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Second case: If π = 2, π(2) = 2π + 3
LHL = limπ₯β2β
π(π₯) = limπ₯β2β
(2π₯ + 3) = 7
RHL = limπ₯β2+
π(π₯) = limπ₯β2+
(2π₯ β 3) = 1
Here, at π₯ = 2, LHL β RHL. Hence, the function π is discontinuous at π₯ = 2.
Third case: If π > 2,
π(π) = 2π β 3 and limπ₯βπ
π(π₯) = limπ₯βπ
(2π₯ β 3) = 2π β 3. Here, limπ₯βπ
π(π₯) = π(π)
Therefore, the function π is continuous for all real numbers greater than 2.
Hence, the function π is discontinuous only at π₯ = 2.
7. Find all points of discontinuity of π,
where π is defined by π(π₯) = {|π₯| + 3, If π₯ β€ β3
β2π₯, If β 3 < π₯ < 36π₯ + 2, If π₯ β₯ 3
Solution:
Let π be any real number. According to question,
π < β3 or π = β3 or β3 < π < 3 or π = 3 or π > 3
First case: If π < β3,
π(π₯) = βπ + 3 and limπ₯βπ
π(π₯) = limπ₯βπ
(βπ₯ + 3) = βπ + 3. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than β3.
Second case: If π = β3, π(β3) = β(β3) + 3 = 6
LHL = limπ₯ββ3β
π(π₯) = limπ₯ββ3β
(βπ₯ + 3) = β(β3) + 3 = 6
RHL = limπ₯ββ3+
π(π₯) = limπ₯ββ3+
(β2π₯) = β2(β3) = 6. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous at π₯ = β3.
Third case: If β3 < π < 3,
π(π) = β2π and limπ₯βπ
π(π₯) = limπ₯βπ
π(β2π₯) = β2π. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous at β3 < π₯ < 3.
Fourth case: If π = 3,
LHL = limπ₯βπβ
π(π₯) = limπ₯βπβ
(β2π₯) = β2π
RHL = limπ₯βπ+
π(π₯) = limπ₯βπ+
(6π₯ + 2) = 6π + 2,
Here, at π₯ = 3, LHL β RHL. Hence, the function π is discontinuous at π₯ = 3.
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Fifth case: If π > 3,
π(π) = 6π + 2 and limπ₯βπ
π(π₯) = limπ₯βπ
(6π₯ + 2) = 6π + 2. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all numbers greater than 3.
Hence, the function π is discontinuous only at π₯ = 3.
8. Find all points of discontinuity of π, where π is defined by π(π₯) = {|π₯|
π₯, If π₯ β 0
0, If π₯ = 0
Solution:
After redefining the function π, we get
π(π₯) =
{
β
π₯
π₯= β1, If π₯ < 0
0, If π₯ = 0π₯
π₯= 1, If π₯ > 0
Let π be any real number. According to question, π < 0 or π = 0 or π > 0.
First case: If π < 0,
π(π) = βπ
π= β1 and lim
π₯βππ(π₯) = lim
π₯βπ(β
π₯
π₯) = β1. Hence, lim
π₯βππ(π₯) = π(π)
Hence, the function π is continuous for all real numbers smaller than 0.
Second case: If, π = 0, π(0) = 0
LHL = limπ₯βπβ
π(π₯) = limπ₯βπβ
(βπ₯
π₯) = β1 and RHL = lim
π₯βπ+π(π₯) = lim
π₯βπ+(π₯
π₯) = 1.
Here, at π₯ = 0, LHL β RHL. Hence, the function π is discontinuous at π₯ = 0.
Third case: If π > 0,
π(π) =π
π= 1 and lim
π₯βππ(π₯) = lim
π₯βπ(π₯
π₯) = 1. Here, lim
π₯βππ(π₯) = π(π)
Hence, the function π is continuous for all real numbers greater than 0.
Therefore, the function π is discontinuous only at π₯ = 0.
9. Find all points of discontinuity of π, where π is defined by π(π₯) = {π₯
|π₯|, If π₯ < 0
β1, If π₯ β₯ 0
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Solution:
Redefining the function, we get
π(π₯) = {
π₯
|π₯|=
π₯
βπ₯= β1, If π₯ < 0
β1, If π₯ β₯ 0
Here, limπ₯βπ
π(π₯) = π(π) = β1, where π is a real number.
Hence, the function π is continuous for all real numbers.
10. Find all points of discontinuity of π, where π is defined by π(π₯) = {π₯ + 1, If β₯ 1
π₯2 + 1, If π₯ < 1
Solution:
Given function is defined by π(π₯) = {π₯ + 1, If β₯ 1
π₯2 + 1, If π₯ < 1
Let π be any real number. According to question, π < 1 or π = 1 or π > 1
First case: If π < 1,
π(π) = π2 + 1 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯2 + 1) = π2 + 1. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers smaller than 1.
Second case: If π = 1, π(1) = 1 + 1 = 2
LHL = limπ₯β1β
π(π₯) = limπ₯β1β
(π₯2 + 1) = 1 + 1 = 2
RHL = limπ₯β1+
π(π₯) = limπ₯β1+
(π₯ + 1) = 1 + 1 = 2,
Here, at π₯ = 1, LHL = RHL = π(1). Hence, the function π is continuous at π₯ = 1.
Third case: If π > 1,
π(π) = π + 1 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯ + 1) = π + 1 . Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers greater than 1.
Therefore, the function π is continuous for all real numbers.
11. Find all points of discontinuity of π, where π is defined by π(π₯) = {π₯3 β 3, If π₯ β€ 2
π₯2 + 1, If π₯ > 2
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Solution:
Given function is defined by π(π₯) = {π₯3 β 3, If π₯ β€ 2
π₯2 + 1, If π₯ > 2
Let π be any real number. According to question, π < 2 or π = 2 or π > 2
First case: If π < 2,
π(π) = π3 β 3 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯3 β 3) = π3 β 3. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 2.
Second case: If π = 2, π(2) = 23 β 3 = 5
LHL = limπ₯β2β
π(π₯) = limπ₯β2β
(π₯3 β 3) = 23 β 3 = 5
RHL = limπ₯β2+
π(π₯) = limπ₯β2+
(π₯2 + 1) = 22 + 1 = 5
Here at π₯ = 2, LHL = RHL = π(2)
Hence, the function π is continuous at π₯ = 2.
Third case: If π > 2,
π(π) = π2 + 1 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯2 + 1) = π2 + 1. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for real numbers greater than 2.
Hence, the function π is continuous for all real numbers.
12. Find all points of discontinuity of π, where π is defined by π(π₯) = {π₯10 β 1, If π₯ β€ 1
π₯2, If π₯ > 1
Solution:
Given function is defined by π(π₯) = {π₯10 β 1, If π₯ β€ 1
π₯2, If π₯ > 1
Let π be any real number. According to question, π < 1 or π = 1 or π > 1
First case: If π < 1,
π(π) = π10 β 1 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯10 β 1) = π10 β 1. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 1.
Second case: If π = 1, π(1) = 110 β 1 = 0
LHL = limπ₯β1β
π(π₯) = limπ₯β1β
(π₯10 β 1) = 0
RHL = limπ₯β1+
π(π₯) = limπ₯β1+
(π₯2) = 1
Here at π₯ = 1, LHL β RHL. Hence, the function π is discontinuous at π₯ = 1.
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Third case: If π > 1,
π(π) = π2 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯2) = π2. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real values greater than 1.
Hence, the function π is discontinuous only at π₯ = 1.
13. Is the function defined by π(π₯) = {π₯ + 5, If π₯ β€ 1 π₯ β 5, If π₯ > 1
a continuous function?
Solution:
Given function is defined by π(π₯) = {π₯ + 5, If π₯ β€ 1 π₯ β 5, If π₯ > 1
Let, π be any real number. According to question, π < 1 or π = 1 or π > 1
First case: If π < 1,
π(π) = π + 5 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯ + 5) = π + 5. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 1.
Second case: If π = 1, π(1) = 1 + 5 = 6
LHL = limπ₯β1β
π(π₯) = limπ₯β1β
(π₯ + 5) = 6
RHL = limπ₯β1+
π(π₯) = limπ₯β1+
(π₯ β 5) = β4,
Here at π₯ = 1, LHL β RHL. Hence, the function π is discontinuous at π₯ = 1.
Third case: If π > 1,
π(π) = π β 5 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯ β 5) = π β 5
Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers greater than 1.
Therefore, the function π is discontinuous only at π₯ = 1.
14. Discuss the continuity of the function π,
where π is defined by π(π₯) = { 3, If 0 β€ π₯ β€ 14, If 1 < π₯ < 3
5, If 3 β€ π₯ β€ 10
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Solution:
Given function is defined by π(π₯) = { 3, If 0 β€ π₯ β€ 14, If 1 < π₯ < 3
5, If 3 β€ π₯ β€ 10
Let π be any real number. According to question, π can be
0 β€ π β€ 1 or π = 1 or 1 < π < 3 or π = 3 or 3 β€ π β€ 10
First case: If 0 β€ π β€ 1,
π(π) = 3 and limπ₯βπ
π(π₯) = limπ₯βπ
(3) = 3. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for 0 β€ π₯ β€ 1.
Second case: If π = 1, π(1) = 3
LHL = limπ₯β1β
π(π₯) = limπ₯β1β
(3) = 3
RHL = limπ₯β1+
π(π₯) = limπ₯β1+
(4) = 4,
Here at π₯ = 1, LHL β RHL. Hence, the function π is discontinuous at π₯ = 1.
Third case: If 1 < π < 3,
π(π) = 4 and limπ₯βπ
π(π₯) = limπ₯βπ
(4) = 4. Here limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for 1 < π₯ < 3.
Fourth case: If π = 3,
LHL = limπ₯β3β
π(π₯) = limπ₯β3β
(4) = 4 and RHL = limπ₯β3+
π(π₯) = limπ₯β3+
(5) = 5,
Here at π₯ = 3, LHL β RHL. Hence, the function π is discontinuous at π₯ = 3.
Fifth case: If 3 β€ π β€ 10,
π(π) = 5 and limπ₯βπ
π(π₯) = limπ₯βπ
(5) = 5. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for 3 β€ π₯ β€ 10.
Hence, the function π is discontinuous only at π₯ = 1 and π₯ = 3.
15. Discuss the continuity of the function π,
where π is defined by π(π₯) = {2π₯, If π₯ < 00, If 0 β€ π₯ β€ 1 4π₯, If π₯ > 1
Solution:
Given function is defined by π(π₯) = {2π₯, If π₯ < 00, If 0 β€ π₯ β€ 1 4π₯, If π₯ > 1
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Let π be any real number. According to question,
π < 0 or π = 0 or 0 β€ π β€ 1 or π = 1 or π > 1
First case: If π < 0,
π(π) = 2π and limπ₯βπ
π(π₯) = limπ₯βπ
(2π₯) = 2π. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 0.
Second case: If π = 0, π(0) = 0
LHL = limπ₯β0β
π(π₯) = limπ₯β0β
(2π₯) = 0
RHL = limπ₯β0+
π(π₯) = limπ₯β0+
(0) = 0. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous at π₯ = 0.
Third case: If 0 β€ π β€ 1,
π(π) = 0 and limπ₯βπ
π(π₯) = limπ₯βπ
(0) = 0. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous at 0 β€ π₯ β€ 1.
Fourth case: If π = 1,
LHL = limπ₯β1β
π(π₯) = limπ₯β1β
(0) = 0
RHL = limπ₯β1+
π(π₯) = limπ₯β1+
(4π₯) = 4,
Here, at π₯ = 1, LHL β RHL.
Hence, the function π is discontinuous at π₯ = 1.
Fifth case: If π > 1,
π(π) = 4π and limπ₯βπ
π(π₯) = limπ₯βπ
(4π₯) = 4π.
Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers greater than 1.
Hence, the function π is discontinuous only at π₯ = 1.
16. Discuss the continuity of the function π,
where π is defined by π(π₯) = {β2, If π₯ β€ β12π₯, If β 1 < π₯ β€ 1 2, If π₯ > 1
Solution:
Given function is defined by π(π₯) = {β2, If π₯ β€ β12π₯, If β 1 < π₯ β€ 1 2, If π₯ > 1
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Let π be any real number
According to question, π < β1 or π = β1 or β1 < π₯ β€ 1 or π = 1 or π > 1
First case: If π < β1,
π(π) = β2 and limπ₯βπ
π(π₯) = limπ₯βπ
(β2) = β2. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than β1.
Second case: If π = β1, π(β1) = β2
LHS = limπ₯ββ1β
π(π₯) = limπ₯ββ1β
(β2) = β2
RHL = limπ₯ββ1+
π(π₯) = limπ₯ββ1+
(2π₯) = β2. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous at π₯ = β1.
Third case: If β1 < π₯ β€ 1,
π(π) = 2π and limπ₯βπ
π(π₯) = limπ₯βπ
(2π₯) = 2π. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous at β1 < π₯ β€ 1.
Fourth case: If π = 1,
LHL = limπ₯β1β
π(π₯) = limπ₯β1β
(2π₯) = 2
RHL = limπ₯β1+
π(π₯) = limπ₯β1+
(2) = 2. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous at π₯ = 1.
Fifth case: If π > 1,
π(π) = 2 and limπ₯βπ
π(π₯) = limπ₯βπ
(2) = 2.
Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers greater than 1.
Therefore, the function π is continuous for all real numbers.
17. Find the relationship between π and π so that the function π defined by
π(π₯) = {ππ₯ + 1, If π₯ β€ 3
ππ₯ + 3, If π₯ > 3 is continuous at π₯ = 3.
Solution:
Given functions is defined by π(π₯) = {ππ₯ + 1, If π₯ β€ 3
ππ₯ + 3, If π₯ > 3
Given that the function is continuous at π₯ = 3. Therefore, LHL = RHL = π(3)
β limπ₯β3β
π(π₯) = limπ₯β3+
π(π₯) = π(3)
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β limπ₯β3β
ππ₯ + 1 = limπ₯β3+
ππ₯ + 3 = 3π + 1
β 3π + 1 = 3π + 3 = 3π + 1
β 3π = 3π + 2 β π = π +2
3
Hence, the relationship between π and π is π = π +2
3
18. For what value of π is the function defined by
π(π₯) = {π(π₯2 β 2π₯), If π₯ β€ 0
4π₯ + 1, If π₯ > 0
Continuous at π₯ = 0? What about continuity at π₯ = 1?
Solution:
Given function is defined as π(π₯) = {π(π₯2 β 2π₯), If π₯ β€ 0
4π₯ + 1, If π₯ > 0
Given that the function is continuous at π₯ = 0. Therefore, LHL = RHL = π(0)
β limπ₯β0β
π(π₯) = limπ₯β0+
π(π₯) = π(0)
β limπ₯β0β
π(π₯2 β 2π₯) = limπ₯β0+
4π₯ + 1 = π[(0)2 β 2(0)]
β π[(0)2 β 2(0)] = 4(0) + 1 = π(0)
β 0. π = 1 β π =1
0
Hence, there is no real value of π for which the given function be continuous.
If π₯ = 1,
π(1) = 4(1) + 1 = 5 and limπ₯β1
π(π₯) = limπ₯β1
4(1) + 1 = 5. Here, limπ₯β1
π(π₯) = π(1)
Therefore, the function π is continuous for all real values of π.
19. Show that the function defined by π(π₯) = π₯ β [π₯] is discontinuous at all integral points.
Here [π₯] denotes the greatest integer less than or equal to π₯.
Solution:
Given function is defined by π(π₯) = π₯ β [π₯]
Let π be any integer
LHL = limπ₯βπβ
π(π₯) = limπ₯βπβ
π₯ β [π₯] = π β (π β 1) = 1
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RHL = limπ₯βπ+
π(π₯) = limπ₯βπ+
π₯ β [π₯] = π β (π) = 0,
Here, at π₯ = π, LHL β RHL.
Therefore, the function π is discontinuous for all integers.
20. Is the function defined by π(π₯) = π₯2 β sinπ₯ + 5 continuous at π₯ = π.
Solution:
Given function is defined by π(π₯) = π₯2 β sinπ₯ + 5,
At π₯ = π, π(π) = π2 β sinπ + 5 = π2 β 0 + 5 = π2 + 5
limπ₯βπ
π(π₯) = limπ₯βπ
π₯2 β sinπ₯ + 5 = π2 β sinπ + 5 = π2 β 0 + 5 = π2 + 5
Here, at π₯ = π, limπ₯βπ
π(π₯) = π(π) = π2 + 5
Therefore, the function π(π₯) is continuous at π₯ = π.
21. Discuss the continuity of the following functions:
(a) π(π₯) = sin π₯ + cos π₯
(b) π(π₯) = sin π₯ β cosπ₯
(c) π(π₯) = sinπ₯. cos π₯
Solution:
Let π(π₯) = sin π₯
Let π be any real number. At π₯ = π, π(π) = sinπ
LHL = limπ₯βπβ
π(π₯) = limπ₯βπβ
sin π₯ = limββ0
sin(π β β) = limββ0
sinπ cosβ β cos π sin β = sin π
RHL = limπ₯βπ+
π(π₯) = limπ₯βπ+
sin π₯ = limββ0
sin(π + β) = limββ0
sin π cos β + cos π sinβ = sin π
Here, at π₯ = π, LHL = RHL = π(π).
Hence, the function π is continuous for all real numbers.
Let β(π₯) = cos π₯
Let π be any real number. π₯ = π, β(π) = cos π
πΏπ»πΏ = limπ₯βπβ
β(π₯) = limπ₯βπβ
cos π₯ = limββ0
cos(π β β) = limββ0
cosπ cos β + sinπ sinβ = cos π
RHL = limπ₯βπ+
β(π₯) = limπ₯βπ+
cos π₯ = limββ0
cos(π + β) = limββ0
cos π cosβ β sinπ sinβ = cos π
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Here, at π₯ = π, LHL = RHL = β(π).
Hence, the function β is continuous for all real numbers.
We know that if π and β are two continuous functions, then the functions π + β, π β β and πβ also be a continuous function.
Therefore, (a) π(π₯) = sinπ₯ + cos π₯ (b) π(π₯) = sin π₯ β cosπ₯ and
(c) π(π₯) = sinπ₯. cos π₯ are continuous functions.
22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Solution:
Let π(π₯) = sin π₯
Let π be any real number. At π₯ = π, π(π) = sinπ
LHL = limπ₯βπβ
π(π₯) = limπ₯βπβ
sin π₯ = limββ0
sin(π β β) = limββ0
sinπ cos β β cos π sin β = sinπ
RHL = limπ₯βπ+
π(π₯) = limπ₯βπ+
sin π₯ = limββ0
sin(π + β) = limββ0
sinπ cosβ + cos π sin β = sin π
Here, at π₯ = π, LHL = RHL = π(π).
Hence, the function π is continuous for all real numbers.
Let β(π₯) = cos π₯
Let π be any real number. At π₯ = π, β(π) = cosπ
πΏπ»πΏ = limπ₯βπβ
β(π₯) = limπ₯βπβ
cos π₯ = limββ0
cos(π β β) = limββ0
cos π cosβ + sinπ sinβ = cos π
RHL = limπ₯βπ+
β(π₯) = limπ₯βπ+
cos π₯ = limββ0
cos(π + β) = limββ0
cos π cosβ β sinπ sinβ = cos π
Here, at π₯ = π, LHL = RHL = β(π).
Hence, the function β is continuous for all real numbers.
We know that if π and β are two continuous functions, then the functions π
β, β β 0,
1
β, β β 0
and 1
π, π β 0 are continuous functions.
Therefore, cosecπ₯ =1
sinπ₯, sin π₯ β 0 is continuous β π₯ β ππ(π β π) is continuous.
Hence, cosec π₯ is continuous except π₯ = ππ(π β π).
sec π₯ =1
cosπ₯, cos π₯ β 0 is continuous. β π₯ β
(2π+1)π
2(π β π) is continuous.
Hence, sec π₯ is continuous except π₯ =(2π+1)π
2(π β π).
cot π₯ =cosπ₯
sinπ₯, sin π₯ β 0 is continuous. β π₯ β ππ(π β π) is continuous.
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Hence, cot π₯ is continuous except π₯ = ππ(π β π).
23. Find all points of discontinuity of π, where π(π₯) = {sinπ₯
π₯, If π₯ < 0
π₯ + 1, If π₯ β₯ 0
Solution:
Given function is defined by π(π₯) = {sinπ₯
π₯, If π₯ < 0
π₯ + 1, If π₯ β₯ 0
Let π be any real number. According to question, π < 0 or π = 0 or π > 0
First case: If π < 0
π(π) =sinπ
π and lim
π₯βππ(π₯) = lim
π₯βπ(sinπ₯
π₯) =
sinπ
π. Here, lim
π₯βππ(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 0.
Second case: If π = 0
π(0) = 0 + 1 = 1
LHL = limπ₯β0β
π(π₯) = limπ₯β0β
(π₯ + 1) = 0 + 1 = 1
RHL = limπ₯β0+
π(π₯) = limπ₯β0+
(π₯ + 1) = 0 + 1 = 1,
Here at π₯ = 0, LHL = RHL = π(0). Hence, the function π is continuous at π₯ = 0.
Third case: If π > 0
π(π) = π + 1 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯ + 1) = π + 1. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers greater than 0.
Therefore, the function π is continuous for all real numbers.
24. Determine if π defined by
π(π₯) = {π₯2 sin
1
π₯, If π₯ β 0
0, If π₯ = 0
is a continuous function?
Solution:
Given function is defined by π(π₯) = {π₯2 sin
1
π₯, If π₯ β 0
0, If π₯ = 0
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Let π be any real number. According to question, π β 0 or π = 0
First case: If π β 0
π(π) = π2 sin1
π and lim
π₯βππ(π₯) = lim
π₯βπ(π₯2 sin
1
π₯) = π2 sin
1
π. Here, lim
π₯βππ(π₯) = π(π)
Hence, the function π is continuous for π β 0.
Second case: If, π = 0, π(0) = 0
LHL = limπ₯β0β
π(π₯) = limπ₯β0β
(π₯2 sin1
π₯) = lim
π₯β0(π₯2 sin
1
π₯)
We know that, β1 β€ sin1
π₯β€ 1, π₯ β 0 β βπ₯2 β€ sin
1
π₯β€ π₯2
β limπ₯β0
(βπ₯2) β€ limπ₯β0
sin1
π₯β€ limπ₯β0
π₯2
β 0 β€ limπ₯β0
sin1
π₯β€ 0 β lim
π₯β0sin
1
π₯= 0 β lim
π₯β0βπ₯2 sin
1
π₯= 0 β lim
π₯β0βπ(π₯) = 0
Similarly, RHL = limπ₯β0+
π(π₯) = limπ₯β0+
(π₯2 sin1
π₯) = lim
π₯β0(π₯2 sin
1
π₯) = 0
Here, at π₯ = 0, LHL = RHL = π(0)
Hence, at π₯ = 0, π is continuous.
Therefore, the function π is continuous for all real numbers.
25. Examine the continuity of π, where π is defined by
π(π₯) = {sin π₯ β cosπ₯ , If π₯ β 0
β1, If π₯ = 0
Solution:
Given function is defined by π(π₯) = {sinπ₯ β cos π₯ , If π₯ β 0
β1, If π₯ = 0
Let π be any real number. According to question, π β 0 or π = 0
First case: If π β 0, π(0) = 0 β 1 = β1
LHL = limπβ0β
π(π₯) = limπβ0β
(sin π₯ β cos π₯) = 0 β 1 = β1
RHL = limπβ0+
π(π₯) = limπβ0+
(sin π₯ β cos π₯) = 0 β 1 = β1
Hence, at π₯ β 0, LHL = RHL = π(π₯)
Hence, the function π is continuous at π₯ β 0.
Second case: If, π = 0, π(π) = β1
and limπ₯βπ
π(π₯) = limπ₯βπ
(β1) = β1 Here, limπ₯βπ
π(π₯) = π(π)
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Hence, the function π is continuous at π₯ = 0
Hence, the function π is continuous for all real numbers.
26. Find the values of π so that the function π is continuous at the indicated point.
π(π₯) = {
π cosπ₯
πβ2π₯, If π₯ β
π
2
3, If π₯ =π
2
at π₯ =π
2
Solution:
Given function is defined by π(π₯) = {
π cosπ₯
πβ2π₯, If π₯ β
π
2
3, If π₯ =π
2
at π₯ =π
2
Given that the function is continuous at π₯ =π
2. Therefore, LHL = RHL = π (
π
2)
β limπ₯β
π2
π(π₯) = limπ₯β
π+
2
π(π₯) = π (π
2)
β limπ₯β
π2
π cos π₯
π β 2π₯= lim
π₯βπ+
2
π cos π₯
π β 2π₯= 3
β limββ0
π cos(π
2ββ)
πβ2(π
2ββ)
= limββ0
π cos(π
2+β)
πβ2(π
2+β)
= 3
β limββ0
π sinβ
2β= lim
ββ0
βπ sinβ
β2β= 3
βπ
2=
π
2= 3 [β΅ lim
ββ0
sinβ
β= 1]
β π = 6
Hence, for π = 6, the given function π is continuous at the indicated point.
27. Find the values of π so that the function π is continuous at the indicated point.
π(π₯) = {ππ₯2, If π₯ β€ 2
3, If π₯ > 2 at π₯ = 2
Solution:
Given function is defined by π(π₯) = {ππ₯2, If π₯ β€ 2
3, If π₯ > 2 at π₯ = 2
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Given that the function is continuous at π₯ = 2.
Therefore, LHL = RHL = π(2)
β limπ₯β2β
π(π₯) = limπ₯β2+
π(π₯) = π(2)
β limπ₯β2β
ππ₯2 = limπ₯β2+
3 = π(2)2
β 4π = 3 = 4π
β π =3
4
Hence, for π =3
4, the given function π is continuous at the indicated point.
28. Find the values of π so that the function π is continuous at the indicated point.
π(π₯) = {ππ₯ + 1, If π₯ β€ π
cosπ₯, If π₯ > π at π₯ = π
Solution:
Given function is π(π₯) = {ππ₯ + 1, If π₯ β€ π
cos π₯, If π₯ > π at π₯ = π
Given that the function is continuous at π₯ = π,
Therefore, LHL = RHL = π(π)
β limπ₯βπβ
π(π₯) = limπ₯βπ+
π(π₯) = π(π)
β limπ₯βπβ
ππ₯ + 1 = limπ₯βπ+
cos π₯ = π(π) + 1
β π(π) + 1 = cos π = ππ + 1
β ππ + 1 = β1 = ππ + 1
β ππ = β2
β π = β2
π
Hence, for π = β2
π, the given function π is continuous at the indicated point.
29. Find the values of π so that the function π is continuous at the indicated point.
π(π₯) = {ππ₯ + 1, If π₯ β€ 53π₯ β 5, If π₯ > 5
at π₯ = 5
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Solution:
Given function is defined by π(π₯) = {ππ₯ + 1, If π₯ β€ 53π₯ β 5, If π₯ > 5
at π₯ = 5
Given that the function is continuous at π₯ = 5.
Therefore, LHL = RHL = π(5)
β limπ₯β5β
π(π₯) = limπ₯β5+
π(π₯) = π(5)
β limπ₯β5β
ππ₯ + 1 = limπ₯β5+
3π₯ β 5 = 5π + 1
β 5π + 1 = 15 β 5 = 5π + 1
β 5π = 9
β π =9
5
Hence, for π =9
5, the given function π is continuous at the indicated point.
30. Find the values of π and π such that the function defined by
π(π₯) = {5, If π₯ β€ 2
ππ₯ + π, If 2 < π₯ < 1021, If π₯ β₯ 10
is continuous function.
Solution:
Given function is π(π₯) = {5, If π₯ β€ 2
ππ₯ + π, If 2 < π₯ < 1021, If π₯ β₯ 10
Given that the function is continuous at π₯ = 2. Therefore, LHL = RHL = π(2)
β limπ₯β2β
π(π₯) = limπ₯β2+
π(π₯) = π(2)
β limπ₯β2β
5 = limπ₯β2+
ππ₯ + π = 5
β 2π + π = 5 β¦(i)
Given that the function is continuous at π₯ = 10. Therefore, LHL = RHL = π(10)
β limπ₯β10β
π(π₯) = limπ₯β10+
π(π₯) = π(10)
β limπ₯β10β
ππ₯ + π = limπ₯β10+
21 = 21
β 10π + π = 21 β¦(ii)
Solving the equation (i) and (ii), we get
π = 2 π = 1
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31. Show that the function defined by π(π₯) = cos(π₯2) is a continuous function.
Solution:
Given function is defined by π(π₯) = cos(π₯2)
Assuming that the functions are well defined for all real numbers, we can write the given function π in the combination of π and β(π = ππβ). Where, π(π₯) = cosπ₯ and β(π₯) = π₯2, if π and β both are continuous function then π also be continuous.
[β΅ ππβ(π₯) = π(β(π₯)) = π(π₯2) = cos(π₯2)]
Let the function π(π₯) be cos π₯
Let π be any real number. At π₯ = π, π(π) = cos π
limπ₯βπ
π(π₯) = limπ₯βπ
cos π₯ = limββ0
cos(π + β) = limββ0
cos π cos β β sinπ sin β = cosπ
Here, limπ₯βπ
π(π₯) = π(π). Hence, t\he function π is continuous for all real numbers.
And let the function β(π₯) be π₯2
Let π be any real number. At π₯ = π, β(π) = π2
limπ₯βπ
β(π₯) = limπ₯βπ
π₯2 = π2
Here, limπ₯βπ
β(π₯) = β(π). Hence, the function β is continuous for all real numbers.
Therefore, π and β both are continuous function. Hence, π is continuous.
32. Show that the function defined by π(π₯) = |cos π₯| is a continuous function.
Solution:
Given that the function is defined by π(π₯) = |cos π₯|
Assuming that the functions are well defined for all real numbers, we can write the given function π in the combination of π and β(π = ππβ). Where, π(π₯) = |π₯| and β(π₯) = cos π₯. If π and β both are continuous function then π also be continuous.
[β΅ ππβ(π₯) = π(β(π₯)) = π(cos π₯) = |cos π₯|]
Let the function π(π₯) be |π₯|
Rearranging the function π, we get
π(π₯) = {βπ₯, If π₯ < 0π₯, If π₯ β₯ 0
Let π be any real number. According to question, π < 0 or π = 0 or π > 0
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First case: If π < 0,
π(π) = 0 and limπ₯βπ
π(π₯) = limπ₯βπ
(βπ₯) = 0, here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 0.
Second case: If π = 0, π(0) = 0 + 1 = 1
LHL = limπ₯β0β
π(π₯) = limπ₯β0β
(βπ₯) = 0
RHL = limπ₯β0+
π(π₯) = limπ₯β0+
(π₯) = 0,
Here at π₯ = 0, LHL = RHL = π(0)
Hence, the function π is continuous at π₯ = 0.
Third case: If π > 0,
π(π) = 0 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯) = 0. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers greater than 0.
Hence, the function π is continuous for all real numbers.
And let the function β(π₯) be cos π₯
Let π be any real number. At π₯ = π, β(π) = cosπ
limπ₯βπ
β(π₯) = limπ₯βπ
cos π₯ = cos π
Here, limπ₯βπ
β(π₯) = β(π). Hence, the function β is continuous for all real numbers.
Therefore, π and β both are continuous function. Hence, π is continuous.
33. Examine that sin|π₯| is a continuous function.
Solution:
Let the given function be π(π₯) = sin|π₯|
Assuming that the functions are well defined for all real numbers, we can write the given function π in the combination of π and β(π = βππ). Where, β(π₯) = sin π₯ and π(π₯) = |π₯|. If π and β both are continuous function then π also be continuous.
[β΅ βππ(π₯) = β(π(π₯)) = β(|π₯|) = sin|π₯|]
Function β(π₯) = sin π₯
Let π be any real number. At π₯ = π, β(π) = sinπ
limπ₯βπ
β(π₯) = limπ₯βπ
sin π₯ = sinπ
Here, limπ₯βπ
β(π₯) = β(π). Hence, the function β is continuous for all real numbers.
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Function π(π₯) = |π₯|
Redefining the function π, we get
π(π₯) = {βπ₯, If π₯ < 0π₯, If π₯ β₯ 0
Let π be any real number. According to question, π < 0 or π = 0 or π > 0
First case: If π < 0,
π(π) = 0 and limπ₯βπ
π(π₯) = limπ₯βπ
(βπ₯) = 0. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 0.
Second case: If π = 0, π(0) = 0 + 1 = 1
LHL = limπ₯β0β
π(π₯) = limπ₯β0β
(βπ₯) = 0
RHL = limπ₯β0+
π(π₯) = limπ₯β0+
(π₯) = 0
Here at π₯ = 0, LHL = RHL = π(0)
Hence at π₯ = 0, the function π is continuous.
Third case: If π > 0,
π(π) = 0 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯) = 0. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers greater than 0.
Hence, the function π is continuous for all real numbers.
Therefore, π and β both are continuous function. Hence, π is continuous.
34. Find all the points of discontinuity of π defined by π(π₯) = |π₯| β |π₯ + 1|.
Solution:
Given that the function is defined by π(π₯) = |π₯| β |π₯ + 1|
Assuming that the functions are well defined for all real numbers, we can write the given function π in the combination of π and β(π = π β β), where, π(π₯) = |π₯| and β(π₯) =|π₯ + 1|. If π and β both are continuous function then π also be continuous.
Function π(π₯) = |π₯|
Redefining the function π, we get,
π(π₯) = {βπ₯, If π₯ < 0π₯, If π₯ β₯ 0
Let π be any real number. According to question, π < 0 or π = 0 or π > 0
First case: If π < 0,
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π(π) = 0 and limπ₯βπ
π(π₯) = limπ₯βπ
(βπ₯) = 0. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers less than 0.
Second case: If π = 0, π(0) = 0 + 1 = 1
LHL = limπ₯β0β
π(π₯) = limπ₯β0β
(βπ₯) = 0 and RHL = limπ₯β0+
π(π₯) = limπ₯β0+
(π₯) = 0
Here, at π₯ = 0, LHL = RHL = g(0)
Hence, the function π is continuous at π₯ = 0.
Third case: If π > 0,
π(π) = 0 and limπ₯βπ
π(π₯) = limπ₯βπ
(π₯) = 0. Here, limπ₯βπ
π(π₯) = π(π)
Hence, the function π is continuous for all real numbers more than 0.
Hence, the function π is continuous for all real numbers.
Function β(π₯) = |π₯ + 1|
Redefining the function β, we get
β(π₯) = {β(π₯ + 1), If π₯ < β1
π₯ + 1, If π₯ β₯ β1
Let π be any real number. According to question, π < β1 or π = β1 or π > β1
First case: If π < β1,
β(π) = β(π + 1) and limπ₯βπ
β(π₯) = limπ₯βπ
β(π + 1) = β(π + 1). Here, limπ₯βπ
β(π₯) = β(π)
Hence, the function π is continuous for all real numbers less than β1.
Second case: If π = β1, β(β1) = β1 + 1 = 0
LHL = limπ₯ββ1β
β(π₯) = limπ₯ββ1β
β(β1 + 1) = 0
RHL = limπ₯ββ1+
β(π₯) = limπ₯ββ1+
(π₯ + 1) = β1 + 1 = 0
Here at π₯ = β1, LHL = RHL = β(β1)
Hence, the function β is continuous at π₯ = β1.
Third case: If π > β1
β(π) = π + 1 and limπ₯βπ
β(π₯) = limπ₯βπ
(π + 1) = π + 1. Here, limπ₯βπ
β(π₯) = β(π)
Hence, the function π is continuous for all real numbers greater than β1.
Hence, the function β is continuous for all real numbers.
Therefore, π and β both are continuous function. Hence, π is continuous.
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Exercise π. π
1. Differentiate the functions with respect to π₯
sin(π₯2 + 5)
Solution:
Let π¦ = sin(π₯2 + 5)
Therefore,
ππ¦
ππ₯= cos(π₯2 + 5).
π
ππ₯(π₯2 + 5)
= cos(π₯2 + 5). 2π₯
Hence, π(sin(π₯2+5))
ππ₯ = cos(π₯2 + 5). 2π₯
2. Differentiate the functions with respect to π₯
cos(sin π₯)
Solution:
Let π¦ = cos(sin π₯)
Therefore,
ππ¦
ππ₯= βsin(sinπ₯).
π
ππ₯(sin π₯)
= βsin(sinπ₯) . cos π₯
Hence, π((cos(sinπ₯)))
ππ₯ = β sin(sinπ₯) . cos π₯.
3. Differentiate the functions with respect to π₯
sin(ππ₯ + π)
Solution:
Let π¦ = sin(ππ₯ + π)
Therefore,
ππ¦
ππ₯= cos(ππ₯ + π).
π
ππ₯(ππ₯ + π)
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= cos(ππ₯ + π). π
Hence, π(sin(ππ₯+π))
ππ₯= cos(ππ₯ + π). π
4. Differentiate the functions with respect to π₯
sec(tan(βπ₯))
Solution:
Let π¦ = sec(tan(βπ₯))
Therefore,
ππ¦
ππ₯= sec(tanβπ₯) tan(tanβπ₯).
π
ππ₯(tanβπ₯)
= sec(tanβπ₯) tan(tanβπ₯). sec2βπ₯π
ππ₯(βπ₯)
= sec(tanβπ₯) tan(tanβπ₯) . sec2βπ₯ (1
2βπ₯)
Hence, π(sec(tan(βπ₯)))
ππ₯= sec(tanβπ₯) tan(tanβπ₯) . sec2βπ₯ (
1
2βπ₯)
5. Differentiate the functions with respect to π₯
sin(ππ₯+π)
cos(ππ₯+π)
Solution:
Let π¦ =sin(ππ₯+π)
cos(ππ₯+π)
Therefore,
ππ¦
ππ₯=cos(ππ₯ + π).
πππ₯sin(ππ₯ + π) β sin(ππ₯ + π).
πππ₯cos(ππ₯ + π)
[cos(ππ₯ + π)]2
=cos(ππ₯ + π). sin(ππ₯ + π).
πππ₯(ππ₯ + π) β sin(ππ₯ + π). [β sin(ππ₯ + π).
πππ₯(ππ₯ + π)]
cos2(ππ₯ + π)
=cos(ππ₯+π).sin(ππ₯+π).π+sin(ππ₯+π).sin(ππ₯+π)π
cos2(ππ₯+π)
Hence, π(
sin(ππ₯+π)
cos(ππ₯+π))
ππ₯=
cos(ππ₯+π).sin(ππ₯+π).π+sin(ππ₯+π).sin(ππ₯+π)π
cos2(ππ₯+π)
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6. Differentiate the functions with respect to π₯
cos π₯3. sin2(π₯5)
Solution:
Let π¦ = cos π₯3. sin2(π₯5)
Therefore,
ππ¦
ππ₯= cos π₯3.
π
ππ₯sin2(π₯5) + sin2(π₯5).
π
ππ₯cosπ₯3
= cos π₯3. 2 sin π₯5 cosπ₯5.π
ππ₯π₯5 + sin2(π₯5)[β sin π₯3].
π
ππ₯π₯3
= cos π₯3. 2 sin π₯5 cosπ₯5. 5π₯4 β sin2(π₯5) sin π₯3. 3π₯2
Hence, π(cosπ₯3.sin2(π₯5))
ππ₯= cos π₯3. 2 sin π₯5 cos π₯5. 5π₯4 β sin2(π₯5) sin π₯3. 3π₯2
7. Differentiate the functions with respect to π₯
2βcot(π₯2)
Solution:
Let π¦ = 2βcot(π₯2)
Therefore,
ππ¦
ππ₯= 2.
1
2βcot(π₯2).π
ππ₯[cot(π₯2)]
=1
βcot(π₯2). [β cossec π₯2].
π
ππ₯π₯2
=1
βcot(π₯2). [β cossec π₯2]. 2π₯
Hence, π(2βcot(π₯2))
ππ₯=
1
βcot(π₯2). [β cossec π₯2]. 2π₯
8. Differentiate the functions with respect to π₯
cos(βπ₯)
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Solution:
Let π¦ = cos(βπ₯)
Therefore,
ππ¦
ππ₯= βsin(βπ₯).
π
ππ₯βπ₯
= βsin(βπ₯).1
2βπ₯
Hence, π(cos(βπ₯))
ππ₯= β sin(βπ₯).
1
2βπ₯
9. Prove that the function π given by π(π₯) = |π₯ β 1|, π₯ β π , is not differentiable at π₯ = 1.
Solution:
At π₯ = 1,
LHD = limββ0
π(1 β β) β π(1)
ββ= lim
ββ0
|1 β β β 1| β |1 β 1|
ββ= limββ0
β
ββ= β1
RHD = limββ0
π(1+β)βπ(1)
β= lim
ββ0
|1+ββ1|β|1β1|
β= lim
ββ0
β
β= 1
Here, LHD β RHD, therefore,
the function π(π₯) = |π₯ β 1|, π₯ β π , is not differentiable at π₯ = 1.
10. Prove that the greatest integer function defined by π(π₯) = [π₯], 0 < π₯ < 3, is not differentiable at π₯ = 1 and π₯ = 2.
Solution:
At π₯ = 1,
LHD = limββ0
π(1 β β) β π(1)
ββ= lim
ββ0
[1 β β] β |1|
ββ= limββ0
0 β 1
ββ= β
RHD = limββ0
π(1 + β) β π(1)
β= lim
ββ0
[1 + β] β [1]
β= limββ0
1 β 1
β= 0
Here, LHD β RHD, therefore,
The function π(π₯) = [π₯], 0 < π₯ < 3, is not differentiable at π₯ = 1.
At π₯ = 2,
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LHD = limββ0
π(1 β β) β π(1)
ββ= lim
ββ0
[2 β β] β [2]
ββ= lim
ββ0
1 β 2
ββ= β
RHD = limββ0
π(1 + β) β π(1)
β= lim
ββ0
[2 + β] β [2]
β= lim
ββ0
2 β 2
β= 0
Here, LHD β RHD, therefore,
The function π(π₯) = [π₯], 0 < π₯ < 3, is not differentiable at π₯ = 2.
Exercise π. π
Find ππ¦
ππ₯ in the following:
1. 2π₯ + 3π¦ = sin π₯
Solution:
Given equation is 2π₯ + 3π¦ = sinπ₯
Differentiating both sides with respect to π₯, we get
π
ππ₯(2π₯) +
π
ππ₯(3π¦) =
π
ππ₯sinπ₯
β 2+ 3ππ¦
ππ₯= cosπ₯
βππ¦
ππ₯=
cosπ₯β2
3
2. 2π₯ + 3π¦ = sin π¦
Solution:
Given equation is 2π₯ + 3π¦ = sinπ¦
Differentiating both sides with respect to π₯, we get
π
ππ₯(2π₯) +
π
ππ₯(3π¦) =
π
ππ₯sinπ¦ β 2 + 3
ππ¦
ππ₯= cos π¦
ππ¦
ππ₯
βππ¦
ππ₯(cosπ¦ β 3) = 2
βππ¦
ππ₯=
2
cosπ¦β3
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3. ππ₯ + ππ¦2 = cos π¦
Solution:
Given equation is ππ₯ + ππ¦2 = cos π¦
Differentiating both sides with respect to π₯, we get
π
ππ₯(ππ₯) +
π
ππ₯(ππ¦2) =
π
ππ₯cos π¦ β π + 2ππ¦
ππ¦
ππ₯= βsinπ¦
ππ¦
ππ₯
βππ¦
ππ₯(2ππ¦ + sinπ¦) = βπ β
ππ¦
ππ₯= β
π
2ππ¦+sinπ¦
4. π₯π¦ + π¦2 = tan π₯ + π¦
Solution:
Given equation is π₯π¦ + π¦2 = tanπ₯ + π¦
Differentiating both sides with respect to π₯, we get
π
ππ₯(π₯π¦) +
π
ππ₯(π¦2) =
π
ππ₯tan π₯ +
ππ¦
ππ₯
β π₯ππ¦
ππ₯+ π¦ + 2π¦
ππ¦
ππ₯= sec2 π₯ +
ππ¦
ππ₯
βππ¦
ππ₯(π₯ + 2π¦ β 1) = sec2 π₯ β π¦
βππ¦
ππ₯=
sec2 π₯βπ¦
π₯+2π¦β1
5. π₯2 + π₯π¦ + π¦2 = 100
Solution:
Given equation is π₯2 + π₯π¦ + π¦2 = 100
Differentiating both sides with respect to π₯, we get
π
ππ₯π₯2 +
π
ππ₯(π₯π¦) +
π
ππ₯π¦2 =
π
ππ₯(100)
β 2π₯ + π₯ππ¦
ππ₯+ π¦ + 2π¦
ππ¦
ππ₯= 0
βππ¦
ππ₯(π₯ + 2π¦) = 2π₯ + π¦ β
ππ¦
ππ₯=
2π₯+π¦
π₯+2π¦
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6. π₯3 + π₯2π¦ + π₯π¦2 + π¦3 = 81
Solution:
Given equation is π₯3 + π₯2π¦ + π₯π¦2 + π¦3 = 81
Differentiating both sides with respect to π₯, we get
π
ππ₯π₯3 +
π
ππ₯(π₯2π¦) +
π
ππ₯(π₯π¦2) +
π
ππ₯π¦3 =
π
ππ₯81
β 3π₯2 + π₯2ππ¦
ππ₯+ π¦. 2π₯ + π₯. 2π¦
ππ¦
ππ₯+ π¦2. 1 + 3π¦2
ππ¦
ππ₯= 0
βππ¦
ππ₯(π₯2 + 2π₯π¦ + 3π¦2) = β(3π₯2 + 2π₯π¦ + π¦2) β
ππ¦
ππ₯= β
3π₯2+2π₯π¦+π¦2
π₯2+2π₯π¦+3π¦2
7. sin2 π¦ + cosπ₯π¦ = π
Solution:
Given equation is sin2 π¦ + cos π₯π¦ = π
Differentiating both sides with respect to π₯, we get
π
ππ₯sin2 π¦ +
π
ππ₯cos π₯π¦ =
π
ππ₯π
β 2sinπ¦ cos π¦ππ¦
ππ₯β sinπ₯π¦ (π₯
ππ¦
ππ₯+ π¦) = 0
β sin2π¦ππ¦
ππ₯β π₯ sinπ₯π¦
ππ¦
ππ₯β π¦ sin π₯π¦ = 0
β (sin2π¦ β π₯ sinπ₯π¦)ππ¦
ππ₯= π¦ sinπ₯π¦
βππ¦
ππ₯=
π¦ sinπ₯π¦
sin2π¦βπ₯ sinπ₯π¦
8. sin2 π₯ + cos2 π¦ = 1
Solution:
Given equation is sin2 π₯ + cos2 π¦ = 1
Differentiating both sides with respect to π₯, we get
π
ππ₯sin2 π₯ +
π
ππ₯cos2 π¦ =
π
ππ₯1
β 2sinπ₯ cos π₯ + 2 cosπ¦ (β sinπ¦)ππ¦
ππ₯= 0
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β sin2π₯ β sin2π¦ππ¦
ππ₯= 0 β
ππ¦
ππ₯=
sin2π₯
sin2π¦
9. π¦ = sinβ1 (2π₯
1+π₯2)
Solution:
Given equation is π¦ = sinβ1 (2π₯
1+π₯2)
Let π₯ = tanπ
Therefore, π¦ = sinβ1 (2 tanπ
1+tan2 π) = sinβ1(sin 2π) = 2π = 2 tanβ1 π₯
β π¦ = 2 tanβ1 π₯
Differentiating both sides with respect to π₯, we get
ππ¦
ππ₯=
2
1+π₯2
10. π¦ = tanβ1 (3π₯βπ₯3
1β3π₯2) , β
1
β3< π₯ <
1
β3
Solution:
Given equation is π¦ = tanβ1 (3π₯βπ₯3
1β3π₯2)
Let π₯ = tanπ
Therefore, π¦ = tanβ1 (3 tanπβtan3 π
1β3 tan2 π)
= tanβ1(tan 3π) = 3π = 3 tanβ1 π₯ β π¦ = 3 tanβ1 π₯
Differentiating both sides with respect to π₯, we get
ππ¦
ππ₯=
3
1+π₯2
11. π¦ = cosβ1 (1βπ₯2
1+π₯2) , 0 < π₯ < 1
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Solution:
Given equation is π¦ = cosβ1 (1βπ₯2
1+π₯2)
Let π₯ = tanπ
Therefore, π¦ = cosβ1 (1βtan2 π
1+tan2 π)
= cosβ1(cos 2π) = 2π = 2 tanβ1 π₯
β π¦ = 2 tanβ1 π₯
Differentiating both sides with respect to π₯, we get
ππ¦
ππ₯=
2
1+π₯2
12. π¦ = sinβ1 (1βπ₯2
1+π₯2) , 0 < π₯ < 1
Solution:
Given equation is π¦ = sinβ1 (1βπ₯2
1+π₯2)
Let π₯ = tanπ
Therefore,
π¦ = sinβ1 (1βtan2 π
1+tan2 π)
= sinβ1(cos 2π) = sinβ1 {sin (π
2β 2π)} =
π
2β 2π =
π
2β 2 tanβ1 π₯
β π¦ =π
2β 2 tanβ1 π₯
Differentiating both sides with respect to π₯, we get
ππ¦
ππ₯= 0 β
2
1+π₯2= β
2
1+π₯2
13. π¦ = cosβ1 (2π₯
1+π₯2) , β1 < π₯ < 1
Solution:
Given equation is π¦ = cosβ1 (2π₯
1+π₯2)
Let π₯ = tanπ
Therefore, π¦ = cosβ1 (2 tanπ
1+tan2 π)
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= cosβ1(sin 2π) = cosβ1 {cos (π
2β 2π)} =
π
2β 2π =
π
2β 2 tanβ1 π₯
β π¦ =π
2β 2 tanβ1 π₯
Differentiating both sides with respect to π₯, we get
ππ¦
ππ₯= 0 β
2
1+π₯2= β
2
1+π₯2
14. π¦ = sinβ1(2π₯β1 β π₯2),β1
β2< π₯ <
1
β2
Solution:
Given equation is π¦ = sinβ1(2π₯β1 β π₯2)
Let π₯ = sin π
Therefore, π¦ = sinβ1(2 sinπβ1 β sin2 π)
= sinβ1(2 sinπ cosπ) = sinβ1(sin2π) = 2π = 2 sinβ1 π₯
β π¦ = 2 sinβ1 π₯
Differentiating both sides with respect to π₯, we get
ππ¦
ππ₯=
2
β1βπ₯2
15. π¦ = secβ1 (1
2π₯2β1) , 0 < π₯ <
1
β2
Solution:
Given equation is π¦ = secβ1 (1
2π₯2β1)
Let π₯ = cosπ
Therefore, π¦ = secβ1 (1
2 cos2 πβ1) = secβ1 (
1
cos2π)
secβ1(sec 2π) = 2π = 2 cosβ1 π₯
β π¦ = 2 cosβ1 π₯
Differentiating both sides with respect to π₯, we get
ππ¦
ππ₯= β
2
β1βπ₯2
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Exercise π. π
1. Differentiate the following w.r.t. π₯:
ππ₯
sinπ₯
Solution:
Given expression is ππ₯
sinπ₯
Let π¦ =ππ₯
sinπ₯ therefore,
ππ¦
ππ₯=
ππ₯.π
ππ₯sinπ₯βsinπ₯
π
ππ₯ππ₯
sin2 π₯=
ππ₯.cosπ₯βsinπ₯.ππ₯
sin2 π₯=
ππ₯(cosπ₯βsinπ₯)
sin2 π₯
2. πsinβ1 π₯
Solution:
Given expression is πsinβ1 π₯
Let π¦ = πsinβ1 π₯ , therefore,
ππ¦
ππ₯= πsin
β1 π₯ .π
ππ₯sinβ1 π₯ = πsin
β1 π₯ .1
β1βπ₯2=
πsinβ1 π₯
β1βπ₯2.
3. ππ₯3
Solution:
Given expression is ππ₯3
Let π¦ = ππ₯3, therefore,
ππ¦
ππ₯= ππ₯
3.π
ππ₯π₯3 = ππ₯
3. 3π₯2 = 3π₯2ππ₯
3
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4. sin(tanβ1 πβπ₯)
Solution:
Given expression is sin(tanβ1 πβπ₯)
Let π¦ = sin(tanβ1 πβπ₯), therefore,
ππ¦
ππ₯= cos(tanβ1 πβπ₯) .
π
ππ₯tanβ1 πβπ₯ = cos(tanβ1 πβπ₯) .
1
1+(πβπ₯)2.π
ππ₯πβπ₯
= cos(tanβ1 πβπ₯) .1
1+πβ2π₯. (βπβπ₯) = β
πβπ₯ cos(tanβ1 πβπ₯)
1+πβ2π₯.
5. log(cos ππ₯)
Solution:
Given expression is log(cos ππ₯)
Let π¦ = log(cos ππ₯),
Therefore,
ππ¦
ππ₯=
1
cosππ₯.π
ππ₯cos ππ₯ =
1
cosππ₯(β sin ππ₯)
π
ππ₯ππ₯ = β tan ππ₯. ππ₯
6. ππ₯ + ππ₯2+β―+ ππ₯
5
Solution:
Given expression is ππ₯ + ππ₯2+β―+ ππ₯
5
Let π¦ = ππ₯ + ππ₯2+ ππ₯
3+ ππ₯
4+ ππ₯
5, therefore,
ππ¦
ππ₯= ππ₯ + ππ₯
2 π
ππ₯π₯2 + ππ₯
3 π
ππ₯π₯3 + ππ₯
4 π
ππ₯π₯4 + ππ₯
5 π
ππ₯π₯5
= ππ₯ + ππ₯2. 2π₯ + ππ₯
3. 3π₯2 + ππ₯
4. 4π₯3 + ππ₯
5. 5π₯4
= ππ₯ + 2π₯ππ₯2+ 3π₯2ππ₯
3+ 4π₯3ππ₯
4+ 5π₯4ππ₯
5
7. βπβπ₯, π₯ > 0
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Solution:
Given expression is βπβπ₯, π₯ > 0
Let π¦ = βπβπ₯
Therefore,
ππ¦
ππ₯=
1
2βπβπ₯
π
ππ₯πβπ₯ =
1
2βπβπ₯. πβπ₯.
π
ππ₯βπ₯ =
1
2βπβπ₯. πβπ₯.
1
2βπ₯=
βπβπ₯
4βπ₯
8. log(log π₯), π₯ > 1
Solution:
Given expression is log(log π₯), π₯ > 1
Let π¦ =ππ₯
sinπ₯
Therefore,
ππ¦
ππ₯=
1
logπ₯.π
ππ₯log π₯ =
1
logπ₯.1
π₯=
1
π₯ logπ₯
9. cosπ₯
logπ₯, π₯ > 0
Solution:
Given expression is cosπ₯
log π₯, π₯ > 0
Let π¦ =cosπ₯
logπ₯
Therefore, ππ¦
ππ₯=
log π₯π
ππ₯cosπ₯βcosπ₯
π
ππ₯logπ₯
(logπ₯)2=
logπ₯.(βsinπ₯)βcosπ₯.1
π₯
(logπ₯)2=
β(π₯ sinπ₯ log π₯+cosπ₯)
π₯(logπ₯)2
10. cos(log π₯ + ππ₯)
Solution:
Given expression is cos(log π₯ + ππ₯)
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Let π¦ = cos(log π₯ + ππ₯)
Therefore,
ππ¦
ππ₯= βsin(log π₯ + ππ₯).
π
ππ₯(log π₯ + ππ₯) = β sin(log π₯ + ππ₯). (
1
π₯+ ππ₯)
Exercise π. π
1. Differentiate the functions given
cos π₯. cos 2π₯. cos 3π₯
Solution:
Given function is cos π₯. cos 2π₯. cos 3π₯
Let π¦ = cos π₯. cos 2π₯. cos 3π₯, taking log on both the sides
log π¦ = log cos π₯ + log cos 2π₯ + log cos3π₯
Therefore,
1
π¦
ππ¦
ππ₯=
1
cosπ₯.π
ππ₯cos π₯ +
1
cos2π₯.π
ππ₯cos 2π₯ +
1
cos3π₯.π
ππ₯cos3π₯
βππ¦
ππ₯= π¦ [
1
cosπ₯. (β sinπ₯) +
1
cos2π₯. (β sin2π₯). 2 +
1
cos3π₯. (β sin 3π₯). 3]
βππ¦
ππ₯= cosπ₯. cos 2π₯. cos 3π₯ [β tan π₯ β 2 tan 2π₯ β 3 tan3π₯]
2. Differentiate the functions given
β(π₯β1)(π₯β2)
(π₯β3)(π₯β4)(π₯β5)
Solution:
Given function is β(π₯β1)(π₯β2)
(π₯β3)(π₯β4)(π₯β5)
Let π¦ = β(π₯β1)(π₯β2)
(π₯β3)(π₯β4)(π₯β5), taking log on both the sides
log π¦ =1
2[log(π₯ β 1) + log(π₯ β 2) β log(π₯ β 3) β log(π₯ β 4) β log(π₯ β 5)]
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Therefore,
1
π¦
ππ¦
ππ₯=
1
2[
1
(π₯β1)+
1
(π₯β2)β
1
(π₯β3)β
1
(π₯β4)β
1
(π₯β5)]
βππ¦
ππ₯=
1
2β
(π₯β1)(π₯β2)
(π₯β3)(π₯β4)(π₯β5)[
1
(π₯β1)+
1
(π₯β2)β
1
(π₯β3)β
1
(π₯β4)β
1
(π₯β5)]
3. Differentiate the functions given
(log π₯)cosπ₯
Solution:
Given function is (log π₯)cosπ₯
Let π¦ = (log π₯)cosπ₯ , taking log on both the sides
log π¦ = log(log π₯)cosπ₯ = cos π₯. log log π₯
Therefore,
1
π¦
ππ¦
ππ₯= cos π₯.
π
ππ₯log log π₯ + log log π₯.
π
ππ₯cos π₯
βππ¦
ππ₯= π¦ [cos π₯.
1
logπ₯.1
π₯+ log log π₯. (β sinπ₯)]
βππ¦
ππ₯= (log π₯)cosπ₯ [
cosπ₯βsinπ₯ log logπ₯
π₯ log π₯]
4. Differentiate the functions given
π₯π₯ β 2sinπ₯
Solution:
Given function is π₯π₯ β 2sinπ₯
Let π’ = π₯π₯ and π£ = 2sinπ₯ therefore, π¦ = π’ β π£
Differentiating with respect to π₯ on both sides
ππ¦
ππ₯=
ππ’
ππ₯βππ£
ππ₯ β¦(i)
Here, π’ = π₯π₯ , taking log on both the sides
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log π’ = π₯ log π₯, therefore,
1
π’
ππ’
ππ₯= π₯.
π
ππ₯log π₯ + log π₯.
π
ππ₯π₯ = π₯.
1
π₯+ log π₯. 1 = 1 + log π₯
ππ’
ππ₯= π’[1 + log π₯] = π₯π₯[1 + log π₯] β¦(ii)
and π£ = 2sinπ₯ , taking log on both the sides
log π£ = sinπ₯ log 2, therefore,
1
π£
ππ£
ππ₯= log2 β
π
ππ₯sin π₯ = log 2 β cos π₯
ππ£
ππ₯= π£[cos π₯ log 2] = 2sinπ₯[cos π₯ log 2] β¦(iii)
Putting the value of ππ’
ππ₯ from (ii) and
ππ£
ππ₯ from (iii) in equation (i), we get
ππ¦
ππ₯= π₯π₯[1 + log π₯] β 2sinπ₯[cos π₯ log 2]
5. Differentiate the functions given
(π₯ + 3)2. (π₯ + 4)3. (π₯ + 5)4
Solution:
Given function is (π₯ + 3)2. (π₯ + 4)3. (π₯ + 5)4
Let π¦ = (π₯ + 3)2. (π₯ + 4)3. (π₯ + 5)4, taking log on both the sides
log π¦ = 2 log(π₯ + 3) + 3 log(π₯ + 4) + 4 log(π₯ + 5)
Therefore,
1
π¦
ππ¦
ππ₯= 2.
1
(π₯+3)+ 3.
1
(π₯+4)+ 4.
1
(π₯+5)
βππ¦
ππ₯= π¦ [
2(π₯ + 4)(π₯ + 5) + 3(π₯ + 3)(π₯ + 5) + 4(π₯ + 3)(π₯ + 4)
(π₯ + 3)(π₯ + 4)(π₯ + 5)]
βππ¦
ππ₯= π¦ [
2(π₯2 + 9π₯ + 20) + 3(π₯2 + 8π₯ + 15) + 4(π₯2 + 7π₯ + 12)
(π₯ + 3)(π₯ + 4)(π₯ + 5)]
βππ¦
ππ₯= (π₯ + 3)2 β (π₯ + 4)3 β (π₯ + 5)4 [
9π₯2+70π₯+133
(π₯+3)(π₯+4)(π₯+5)]
βππ¦
ππ₯= (π₯ + 3) β (π₯ + 4)2 β (π₯ + 5)3(9π₯2 + 70π₯ + 133)
6. Differentiate the functions given
(π₯ +1
π₯)π₯+ π₯
(1+1
π₯)
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Solution:
Given function is (π₯ +1
π₯)π₯+ π₯
(1+1
π₯)
Let π’ = (π₯ +1
π₯)π₯
and π£ = π₯(1+
1
π₯), therefore, π¦ = π’ + π£
Differentiating with respect to π₯, we get
ππ¦
ππ₯=
ππ’
ππ₯+ππ£
ππ₯ β¦(i)
Here, π’ = (π₯ +1
π₯)π₯, taking log on both the sides
log π’ = π₯ log (π₯ +1
π₯), therefore,
1
π’
ππ’
ππ₯= π₯ β
π
ππ₯log (π₯ +
1
π₯) + log (π₯ +
1
π₯) β
π
ππ₯π₯
= π₯ β 1
(π₯+1
π₯)β (1 β
1
π₯2) + log (π₯ +
1
π₯) β 1 =
π₯2
π₯2+1β π₯2β1
π₯2+ log (π₯ +
1
π₯)
ππ’
ππ₯= (π₯ +
1
π₯)π₯[π₯2β1
π₯2+1+ log (π₯ +
1
π₯)] β¦(ii)
and π£ = π₯(1+
1
π₯), taking log on both the sides
log π£ = (1 +1
π₯) log π₯, therefore,
1
π£
ππ£
ππ₯= (1 +
1
π₯) Β·
π
ππ₯log π₯ + log π₯ Β·
π
ππ₯(1 +
1
π₯) = (1 +
1
π₯) Β·
1
π₯+ log π₯ Β· (β
1
π₯2)
ππ£
ππ₯= π£ [(
π₯2+1
π₯) Β·
1
π₯βlogπ₯
π₯2] = π₯
(1+1
π₯)[π₯2+1βlogπ₯
π₯2] β¦(iii)
Putting the value of ππ’
ππ₯ from (ii) and
ππ£
ππ₯ from (iii) in equation (i), we get
ππ¦
ππ₯= (π₯ +
1
π₯)π₯[π₯2β1
π₯2+1+ log (π₯ +
1
π₯)] + π₯
(1+1
π₯)[π₯2+1βlogπ₯
π₯2]
7. Differentiate the functions given
(log π₯)π₯ + π₯logπ₯
Solution:
Given function is (log π₯)π₯ + π₯log π₯
Let π’ = (log π₯)π₯ and π£ = π₯logπ₯ , therefore, π¦ = π’ + π£
Differentiating with respect to π₯, we get
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ππ¦
ππ₯=
ππ’
ππ₯+ππ£
ππ₯ β¦(i)
Here, π’ = (log π₯)π₯, taking log on both the sides
log π’ = π₯ log log π₯, therefore,
1
π’
ππ’
ππ₯= π₯.
π
ππ₯log log π₯ + log log π₯.
π
ππ₯π₯
= π₯.1
log π₯.1
π₯+ log log π₯. 1 =
1
log π₯+ log log π₯
ππ’
ππ₯= (log π₯)π₯ [
1 + log π₯. log log π₯
logπ₯]
= (log π₯)π₯β1(1 + log π₯. log log π₯) β¦(ii)
and, π£ = π₯logπ₯ , taking log on both the sides
log π£ = log π₯ log π₯, therefore,
1
π£
ππ£
ππ₯= log π₯.
π
ππ₯log π₯ + ππππ₯ .
π
ππ₯log π₯
= log π₯.1
π₯+ log π₯.
1
π₯
ππ£
ππ₯= π£ [
2 log π₯
π₯] = π₯logπ₯ [
2 log π₯
π₯] = π₯logπ₯β1(2 log π₯) β¦(iii)
Putting the value of ππ’
ππ₯ from (ii) and
ππ£
ππ₯ from (iii) in equation (i), we get
ππ¦
ππ₯= (log π₯)π₯β1(1 + log π₯. log log π₯) + π₯logπ₯β1(2 log π₯)
8. Differentiate the functions given
(sin π₯)π₯ + sinβ1βπ₯
Solution:
Given function is (sin π₯)π₯ + sinβ1 βπ₯
Let π’ = (sinπ₯)π₯ and π£ = sinβ1 βπ₯, therefore, π¦ = π’ + π£
Differentiating with respect to π₯, we get
ππ¦
ππ₯=
ππ’
ππ₯+ππ£
ππ₯ β¦(i)
Here, π’ = (sin π₯)π₯, taking log on both the sides
log π’ = π₯ log sin π₯, therefore,
1
π’
ππ’
ππ₯= π₯.
π
ππ₯log sin π₯ + log sinπ₯.
π
ππ₯π₯
= π₯.1
sin π₯. cos π₯ + log sin π₯. 1 = π₯ cot π₯ + log sin π₯
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ππ’
ππ₯= (sin π₯)π₯(π₯ cot π₯ + log sinπ₯) β¦(ii)
and, π£ = sinβ1 βπ₯, therefore,
1
π£
ππ£
ππ₯= log π₯.
π
ππ₯log π₯ + log π₯.
π
ππ₯log π₯
= log π₯.1
π₯+ log π₯.
1
π₯
ππ£
ππ₯=
1
β1βπ₯.π
ππ₯βπ₯ =
1
β1βπ₯.1
2βπ₯=
1
2βπ₯βπ₯2 β¦(iii)
Putting the value of ππ’
ππ₯ from (ii) and
ππ£
ππ₯ from (iii) in equation (i), we get
ππ¦
ππ₯= (sinπ₯)π₯(π₯ cot π₯ + log sinπ₯) +
1
2βπ₯βπ₯2
9. Differentiate the functions given
π₯sinπ₯ + (sin π₯)cosπ₯
Solution:
Given function is π₯sinπ₯ + (sinπ₯)cosπ₯
Let π’ = π₯sinπ₯ and π£ = (sin π₯)cosπ₯ therefore, π¦ = π’ + π£
Differentiating with respect to π₯, we get
ππ¦
ππ₯=
ππ’
ππ₯+ππ£
ππ₯ β¦(i)
Here, π’ = π₯sinπ₯ , taking log on both the sides
log π’ = sin π₯ log π₯, therefore,
1
π’
ππ’
ππ₯= sinπ₯.
π
ππ₯log π₯ + log π₯.
π
ππ₯sin π₯ = sinπ₯.
1
π₯+ log π₯. cos π₯ =
sinπ₯
π₯+ log π₯ cos π₯
ππ’
ππ₯= π₯sinπ₯ [
sinπ₯
π₯+ log π₯ cosπ₯] = π₯sinπ₯β1(sin π₯ + π₯ log π₯ cos π₯) β¦(ii)
and π£ = (sinπ₯)cosπ₯ , taking log on both the sides
log π£ = cos π₯ log sin π₯, therefore,
1
π£
ππ£
ππ₯= cos π₯.
π
ππ₯log sin π₯ + log sin π₯.
π
ππ₯cosπ₯ = cos π₯.
1
sinπ₯cos π₯ + log sinπ₯(β sin π₯)
ππ£
ππ₯= π£[cos π₯ cot π₯ β sin π₯ log sinπ₯] = (sin π₯)cosπ₯(cos π₯ cot π₯ β sinπ₯ log sin π₯) β¦(iii)
Putting the value of ππ’
ππ₯ from (ii) and
ππ£
ππ₯ from (ii) in equation (i), we get
ππ¦
ππ₯= π₯sinπ₯β1(sin π₯ + π₯ log π₯ cos π₯) + (sin π₯)cosπ₯(cos π₯ cot π₯ β sinπ₯ log sin π₯)
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10. Differentiate the functions given
π₯π₯ cosπ₯ +π₯2+1
π₯2β1
Solution:
Given function is π₯π₯ cosπ₯ +π₯2+1
π₯2β1
Let π’ = π₯π₯ cosπ₯ and π£ =π₯2+1
π₯2β1 therefore, π¦ = π’ + π£
Differentiating with respect to π₯, we get
ππ¦
ππ₯=
ππ’
ππ₯+ππ£
ππ₯ β¦(i)
Here, π’ = π₯π₯ cosπ₯ , taking log on both the sides
log π’ = π₯ log π₯, therefore,
1
π’
ππ’
ππ₯= π₯ cos π₯.
π
ππ₯log π₯ + log π₯.
π
ππ₯π₯ cos π₯ = π₯ cosπ₯.
1
π₯+ log π₯. (βπ₯. sin π₯ + cos π₯)
= cos π₯ β π₯ sin π₯ log π₯ + cos π₯ log π₯
ππ’
ππ₯= π’[cos π₯ β π₯ sinπ₯ log π₯ + cos π₯ log π₯]
= π₯π₯ cosπ₯[cos π₯ β π₯ sin π₯ log π₯ + cosπ₯ log π₯] β¦(ii)
and π£ =π₯2+1
π₯2β1, taking log on both the sides
log π£ = log(π₯2 + 1) β log(π₯2 β 1), therefore,
1
π£
ππ£
ππ₯=
1
π₯2+1Β· 2π₯ β
1
π₯2β1Β· 2π₯ =
2π₯(π₯2β1)β2π₯(π₯2+1)
(π₯2+1)(π₯2β1)=
β4π₯
(π₯2+1)(π₯2β1)
ππ£
ππ₯= π£ [
β4π₯
(π₯2+1)(π₯2β1)] =
π₯2+1
π₯2β1[
β4π₯
(π₯2+1)(π₯2β1)] = β
4π₯
(π₯2β1)2 β¦(iii)
Putting the value of ππ’
ππ₯ from (ii) and
ππ£
ππ₯ from (iii) in equation (i), we get
ππ¦
ππ₯= π₯π₯ cosπ₯[cos π₯ β π₯ sin π₯ log π₯ + cosπ₯ log π₯] β
4π₯
(π₯2β1)2
11. Differentiate the functions given
(π₯ cos π₯)π₯ + (π₯ sin π₯)1
π₯
Solution:
Given function is (π₯ cos π₯)π₯ + (π₯ sinπ₯)1
π₯
Let π’ = (π₯ cos π₯)π₯ and π£ = (π₯ sinπ₯)1
π₯, therefore, π¦ = π’ + π£
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Differentiating with respect to π₯, we get
ππ¦
ππ₯=
ππ’
ππ₯+ππ£
ππ₯ β¦(i)
Here, π’ = (π₯ cos π₯)π₯ , taking log on both the sides
log π’ = π₯ log(π₯ cos π₯), therefore,
1
π’
ππ’
ππ₯= π₯.
π
ππ₯log(π₯ cosπ₯) + log(π₯ cos π₯).
π
ππ₯π₯
= π₯.1
(π₯ cos π₯)(βπ₯ sin π₯ + cosπ₯) + log(π₯ cos π₯). 1 = βπ₯ tan π₯ + 1 + log(π₯ cos π₯)
ππ’
ππ₯= (π₯ cos π₯)π₯[1 β π₯ tan π₯ + log(π₯ cos π₯)]
= (π₯ cos π₯)π₯[1 β π₯ tan π₯ + log(π₯ cos π₯)] β¦(ii)
and, π£ = (π₯ sin π₯)1
π₯, taking log on both the sides
log π£ =1
π₯log(π₯ sin π₯) , therefore,
1
π£
ππ£
ππ₯=1
π₯.π
ππ₯log(π₯ sin π₯) + log(π₯ sin π₯).
π
ππ₯
1
π₯
=1
π₯.
1
π₯ sinπ₯(π₯ cos π₯ + sinπ₯) + log(π₯ sinπ₯) (β
1
π₯2)
ππ£
ππ₯= π£ [
π₯ cot π₯ + 1 β log(π₯ sinπ₯)
π₯2]
= (π₯ sinπ₯)1
π₯ [π₯ cotπ₯+1βlog(π₯ sinπ₯)
π₯2] β¦(iii)
Putting the value of ππ’
ππ₯ from (ii) and
ππ£
ππ₯ from (iii) in equation (i), we get
ππ¦
ππ₯= (π₯ cosπ₯)π₯[1 β π₯ tan π₯ + log(π₯ cos π₯)] + (π₯ sin π₯)
1
π₯ [π₯ cotπ₯+1βlog(π₯ sinπ₯)
π₯2]
12. Find ππ¦
ππ₯ of the functions given
π₯π¦ + π¦π₯ = 1
Solution:
Given function is π₯π¦ + π¦π₯