Derivatives of Inverse Trigonometric Function

[Pages:9]Derivatives of Inverse Trigonometric Function

Inverse Trig Functions

Some trig functions domains' have to be restricted in order for them to have an inverse function ? why? Only functions that are 1-to-1 can have inverse functions

( ) Find dy If y = sin-1 x3 dx

dy = 1 3x2

( ) ( ) dx 1- x3 2

therefore

dy = (3x2 ) ( ) dx 1- x3 2

One more example

( ) Find dy if y = sec-1 ex dx

dy =

1

ex

( ) ( ) dx

ex

ex

2

-1

dy = 1 dx e2x -1

Differentiability of Inverse Functions

If f(x) is differentiable on an interval I, one may wonder whether f-1(x) is also differentiable? The answer to this question hinges on f'(x) being equal to 0 or not . Indeed, if for any , then f-1(x) is also differentiable. Moreover we have

Using Leibniz's notation, the above formula becomes

which is easy to remember.

Example:

Confirm Differentiability of Inverse Function formula for the

function f (x) = x3 +1

Solution:

y = x3 +1 x = y3 +1 y = 3 x -1 x = 3 y -1 f -1(y) = 3 y -1

dy dx

=

d dx

!"

x

3

+1#$

=

3x

2

and

dx dy

=

d dy

"# 3

y

-1$%

=

d dy

"#&(y

-

1

1)3

$ %'

=

13 ( y

)-1

-2 3

( ) dy = 3

dx

3

y -1

2

2

= 3(y -1)3

=

1 dx

dy

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