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

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

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

Example 3

Ify=

(

2 x− 1

x^2

) 3

, find

dy

dx

.

Using the chain rule, letu=

(

2 x− 1

x^2

)

. Then
dy
dx

= 3

(

2 x− 1

x^2

) 2

d

dx

(

2 x− 1

x^2

)

.

To find

d

dx

(

2 x− 1

x^2

)

, use the quotient rule.

Thus,

d

dx

(

2 x− 1

x^2

)

=

(2)(x^2 )−(2x)(2x−1)

(x^2 )^2

=

− 2 x^2 + 2 x

x^4

. Substituting this quantity back

into

dy

dx

= 3

(

2 x− 1

x^2

) 2

d

dx

(

2 x− 1

x^2

)

= 3

(

2 x− 1

x^2

) 2

− 2 x^2 + 2 x

x^4

=

−6(x−1)(2x−1)

x^7

2

.

An alternate solution is to use the product rule and rewritey=

(

2 x− 1

x^2

) 3

as

y =

(2x−1)^3

(x^2 )^3

=

(2x−1)^3

x^6

and use the quotient rule.

Another approach is to expressy =(2x−1)^3 (x−^6 ) and use the product rule. Of course, you

can always use your calculator if you are permitted to do so.

7.2 Derivatives of Trigonometric, Inverse Trigonometric,

Exponential, and Logarithmic Functions

Main Concepts:Derivatives of Trigonometric Functions, Derivatives of Inverse

Trigonometric Functions, Derivatives of Exponential and

Logarithmic Functions

Derivatives of Trigonometric Functions

Summary of Derivatives of Trigonometric Functions

d

dx

(sinx)=cosx

d

dx

(cosx)=−sinx

d

dx

(tanx)=sec^2 x

d

dx

(cotx)=−csc^2 x

d

dx

(secx)=secxtanx

d

dx

(cscx)=−cscxcotx

Note that the derivatives ofcosine,cotangent, andcosecantall have a negative sign.

Example 1

Ify= 6 x^2 +3 secx, find

dy

dx

.

dy

dx

= 12 x+3 secxtanx.