5 Steps to a 5 AP Calculus AB 2019 - William Ma

(Marvins-Underground-K-12) #1
MA 3972-MA-Book May 9, 2018 10:9

114 STEP 4. Review the Knowledge You Need to Score High


Example 4
Using a calculator, findf′(x) and f′(3) if f(x)=

1



x

.


There are several ways of finding f′(x) and f′(9) using a calculator. One way is to use
thed[Differentiate] function. Go to the Home screen. Select F3-Calcand then select
d[Differentiate]. Enterd(1/


(x), x). The result is f′(x)=

− 1


2 x

32. To find f′(3), enter
d(1/


(x), x)|x=3. The result is f′(3)=

− 1


54


.


The Sum, Difference, Product, and Quotient Rules
Ifuandvare two differentiable functions, then
d
dx
(u±v)=

du
dx

±


dv
dx
Sum and Difference Rules
d
dx

(uv)=v
du
dx

+u
dv
dx

Product Rule

d
dx

(u
v

)
=

v
du
dx

−u
dv
dx
v^2
,v= 0 Quotient Rule

Summary of Sum, Difference, Product, and Quotient Rules
(u±v)′=u′±v′ (uv)′=u′v+v′u

(u
v

)′
=
u′v−v′u
v^2
Example 1
Findf′(x)iff(x)=x^3 − 10 x+5.
Using the sum and difference rules, you can differentiate each term and obtain f′(x)=
3 x^2 − 10. Or using your calculator, select the d[Differentiate] function and enter
d(x^3 − 10 x+5,x) and obtain 3x^2 −10.

Example 2
Ify=(3x−5)(x^4 + 8 x−1), find
dy
dx

.


Using the product rule
d
dx
(uv)=v
du
dx
+u
dv
dx
, letu=(3x−5) andv=(x^4 + 8 x−1).

Then
dy
dx
=(3)(x^4 + 8 x−1)+(4x^3 +8)(3x−5)=(3x^4 + 24 x−3)+(12x^4 − 20 x^3 +
24 x−40)= 15 x^4 − 20 x^3 + 48 x−43. Or you can use your calculator and enterd((3x−5)
(x^4 + 8 x−1),x) and obtain the same result.

Example 3
Iff(x)=
2 x− 1
x+ 5
, find f′(x).

Using the quotient rule

(u
v

)′
=
u′v−v′u
v^2
, letu = 2 x −1 and v = x +5. Then

f′(x)=
(2)(x+5)−(1)(2x−1)
(x+5)^2

=


2 x+ 10 − 2 x+ 1
(x+5)^2

=


11


(x+5)^2
,x=− 5 .Or you can use
your calculator and enterd((2x−1)/(x+5),x) and obtain the same result.
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