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

MA 3972-MA-Book May 9, 2018 10:9

Differentiation 121

Example 8

y=ln(lnx)

Letu=lnx. Then

dy

dx

=

1

lnx

du

dx

=

1

lnx

(

1

x

)

=

1

xlnx

.

Example 9

y=log 5 (2x+1)

Letu= 2 x+1. Then

dy

dx

=

1

(2x+1) ln 5

du

dx

=

1

(2x+1) ln 5

·(2)=

2

(2x+1) ln 5

.

Example 10

Write an equation of the line tangent to the curve ofy=exatx=1.

The slope of the tangent to the curvey=exatx=1 is equivalent to the value of the

derivative ofy=exevaluated atx=1. Using your calculator, enterd(e∧(x), x)|x=1 and

obtaine. Thus,m=e, the slope of the tangent to the curve atx=1. Atx=1,y=e^1 =e, and

thus, the point on the curve is (1,e). Therefore, the equation of the tangent isy−e=e(x−1)

ory=ex. (See Figure 7.2-1.)

[–1, 3] by [–2, 8]

Figure 7.2-1

TIP • Never leave a multiple-choice question blank. There is no penalty for incorrect answers.

7.3 Implicit Differentiation

Main Concept:Procedure for Implicit Differentiation

Procedure for Implicit Differentiation

Given an equation containing the variablesxandyfor which you cannot easily solve fory

STRATEGY

in terms ofx, you can find

dy

dx

by doing the following:

Steps

1. Differentiate each term of the equation with respect tox.


2. Move all terms containing
   dy
   dx
   to the left side of the equation and all other terms to the
   right side.


3. Factor out
   dy
   dx
   on the left side of the equation.


4. Solve for
   dy
   dx

.