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Warm-Up
Find a and b such that f(x) is continuous on the entire real line.
3,2
31,
1,2
x
xbax
x
xf
Calculus Warm-Up
• For a function to be differentiable at x=c, what conditions must be met?
1) f must be continuous at c
2) The derivative to the left and
right of c must be equal
Calculus HWQ
• Is this function continuous as x = 2?
• Is the function differentiable at x = 2?
yes
no
2.2 Basic Differentiation Rules 2015
Colorado National MonumentGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003
Recall from last lesson: To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
corner cusp
vertical tangent discontinuity
f x x 2
3f x x
3f x x 1, 0
1, 0
xf x
x
Continuous & DifferentiableContinuous but NOT differentiable everywhere:
**Remember: differentiablility implies continuity but continuity does NOT imply differentiability**
Continuous AND differentiable everywhere :
You try…
At point c, decide whether these graphs are:
a. Continuous
b. Differentiable
c. Both
d. Neither
e)
If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
0dc
dx
example: 3y
0y
The derivative of a constant is zero.
If we find derivatives with the difference quotient:
2 22
0limh
x h xdx
dx h
2 2 2
0
2limh
x xh h x
h
2x
3 33
0limh
x h xdx
dx h
3 2 2 3 3
0
3 3limh
x x h xh h x
h
23x
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
(Pascal’s Triangle)
2
4dx
dx
4 3 2 2 3 4 4
0
4 6 4limh
x x h x h xh h x
h
34x
2 3
We observe a pattern: 2x 23x 34x 45x 56x …
1n ndx nx
dx
examples:
4f x x
34f x x
8y x
78y x
power rule
We observe a pattern: 2x 23x 34x 45x 56x …
d ducu c
dx dx
examples:
1n ndcx cnx
dx
constant multiple rule:
57dx
dx
When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.
4 47 5 35x x
(Each term is treated separately)
d ducu c
dx dx
constant multiple rule:
sum and difference rules:
d du dvu v
dx dx dx d du dv
u vdx dx dx
4 12y x x 34 12y x
4 22 2y x x
34 4dy
x xdx
Example:Find the horizontal tangents of: 4 22 2y x x
34 4dy
x xdx
Horizontal tangents occur when slope = zero.34 4 0x x
3 0x x
2 1 0x x
1 1 0x x x
0, 1, 1x
Plugging the x values into the original equation, we get:
2, 1, 1y y y
(The function is even, so we only get two horizontal tangents.)
Function Rewrite Differentiate Simplify
3
5
2y
x 35
2y x
415'
2y x
4
15'
2y
x
3
5
2y
x 35
8y x
415'
8y x
4
15'
8y
x
2
7
3y
x 27
3y x
14'
3y x
2
7
3y
x 263y x ' 126y x
Often it is helpful to rewrite the function before differentiating:
2
0
2
Consider the function siny
We could make a graph of the slope: slope
1
0
1
0
1Now we connect the dots!
The resulting curve is a cosine curve.
sin cosd
x xdx
2
0
2
We can do the same thing for cosy slope
0
1
0
1
0The resulting curve is a sine curve that has been reflected about the x-axis.
cos sind
x xdx
Higher Order Derivatives:
dyy
dx is the first derivative of y with respect to x.
2
2
dy d dy d yy
dx dx dx dx
is the second derivative.
(y double prime)
dyy
dx
is the third derivative.
4 dy y
dx is the fourth derivative.
We will learn later what these higher order derivatives are used for.
Homework:
Pg.115 Larson Calculus 8th ed. : 31-63 odd, 75, 89
http://college.cengage.com/mathematics/blackboard/content/larson/calc8e/calc8e_solution_main.html?CH=07&SECT=a&TYPE=se