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

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.