104
Derivatives of Algebraic FunctionsPrepared by: Midori Kobayashi
Humber College
27
1(In presentation mode, click on the computer image at the
bottom right for a direct web link to an interesting Wikipedia Math
Site).
CASE STUDY
27.1 - Limits
27.1-EXAMPLE 3-Page 788
727.1-EXAMPLE 3-Page 788-Continued
3 27.1-EXAMPLE 3-Page 788-Continued27.1-EXAMPLE 5-Page 789
The highest power of x
27.1-EXAMPLE 5-Page 789-Continued
27.1-EXAMPLE 6-Page 790
Limit may exist!
27.1-EXAMPLE 6-Page 790-Continued
27.2 The Derivative
27.2-EXAMPLE 13-Page 798
f(x)=x2
Expand using (A+B)2= A2 +2AB + B2
(cont)27.2-EXAMPLE 13-Page 798-Continued
Factored by x
x cancelled
27.2-EXAMPLE 14-Page 798
Substitute x+x into y = 3x2
(cont)27.2-EXAMPLE 14-Page 798-Continued
Note: y = 3x2
(cont)27.2-EXAMPLE 14-Page 798-Continued
Divide by x
(cont)
Let x approachs zero.
27.2-EXAMPLE 14-Page 798-Continued(cont)
27.2-EXAMPLE 14-Page 798-Continued27.2-EXAMPLE 15-Page 800
Substitute x +x into x
(cont)
27.2-EXAMPLE 15-Page 800-Continued(cont)27.2-EXAMPLE 15-Page
800-Continued
(cont)
27.2-EXAMPLE 15-Page 800-Continued 27.3 Rules for
Derivatives27.3-EXAMPLE 21-Page 805
22 is a constant
27.3-EXAMPLE 22-Page 806
(cont)27.3-EXAMPLE 22-Page 806-Continued
27.3-EXAMPLE 22-Page 806-Continued
27.3-EXAMPLE 27-Page 808
27.4 Derivative of a Function Raised to a Power
27.4-EXAMPLE 30-Page 812
Dont forget!
27.4-EXAMPLE 32-Page 812
Dont forget!
27.4-EXAMPLE 33-Page 813
Dont forget!
(cont)27.4-EXAMPLE 33-Page 813-Continued
27.5 Derivatives of Products and Quotients
27.5-EXAMPLE 34-Page 815
uvSou= 2xv=1
27.5-EXAMPLE 35-Page 815
uvSou= 1v= (x-3)-
3627.5-EXAMPLE 38-Page 817
uvSou= 6x2v= 4
27.5-EXAMPLE 40-Page 818
uvSou= 2(t3-3)(3t2)v= (t+1)
(cont)27.5-EXAMPLE 40-Page 818-Continued
27.6 Derivatives of Implicit Relations
27.6-EXAMPLE 43-Page 820
127.6-EXAMPLE 45-Page 821
1
127.6-EXAMPLE 48-Page 822
11Product Rule1yyyProduct Rule
(cont)27.6-EXAMPLE 48-Page 822-Continued
27.6-EXAMPLE 49-Page 823
Dont forget to place ( )27.7 Higher-Order Derivatives
27.7-EXAMPLE 51-Page 825
First derivativeSecond derivative
Third derivativeFourth derivativeFifth derivative27.7-EXAMPLE
52-Page 825
uvSou= 1v= (x-3)-
(cont)27.7-EXAMPLE 52-Page 825-Continued
uvSou= 1v= -(x-3)-3/2
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