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

Differentiation 119

Example 2

Findf′(x)iff(x)=tan−^1

√

x.

Letu=

√

x.Then f′(x)=

1

1 +(

√

x)^2

du

dx

=

1

1 +x

(

1

2

x−

(^12)
)

1

1 +x

(

1

2

√

x

)

=

1

2

√

x(1+x)

.

Example 3

Ify=sec−^1 (3x^2 ), find

dy

dx

.

Letu= 3 x^2. Then

dy

dx

=

1

| 3 x^2 |

√

(3x^2 )^2 − 1

du

dx

=

1

3 x^2

√

9 x^4 − 1

(6x)=

2

x

√

9 x^4 − 1

.

Example 4

Ify=cos−^1

(

1

x

)

, find

dy

dx

.

Letu=

(

1

x

)

. Then
dy
dx

=

− 1

√

1 −

(

1

x

) 2

du

dx

.

Rewriteu=

(

1

x

)

asu=x−^1. Then

du

dx

=− 1 x−^2 =

− 1

x^2

.

Therefore,
dy
dx

=

− 1

√

1 −

(

1

x

) 2

du

dx

=

− 1

√

1 −

(

1

x

) 2

− 1

x^2

=

1

√

x^2 − 1

x^2

(x^2 )

=

1

√

x^2 − 1

|x|

(x^2 )

=

1

|x|

√

x^2 − 1

.

Note that for all of the above exercises, you can find the derivatives by using a calculator,

provided that you are permitted to do so.

Derivatives of Exponential and Logarithmic Functions

Summary of Derivatives of Exponential and Logarithmic Functions

Letube a differentiable function ofx, then

d

dx

(eu)=eu

du

dx

d

dx

(au) =aulna

du

dx

, a>0&a= 1

d

dx

(lnu)=

1

u

du

dx

, u> 0

d

dx

(logau)=

1

ulna

du

dx

, a>0&a= 1.

For the following examples, find

dy

dx

and verify your result with a calculator.