1. Derivatives of Inverse
Trigonometric Function

2. 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

5. Find
dy
dx
If
y = sin −1 ( x 3 )
dy
1
=
×( 3x 2 )
dx
3 2
1− ( x )
therefore
3x 2 )
(
dy
=
dx
3 2
1− ( x )

6. One more example
Find
dy
dx
dy
=
dx e x
y = sec−1 ( e x )
if
1
(e )
x 2
−1
dy
1
=
dx
e 2 x −1
×( e x )

7. 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.

8. Example:
Confirm Differentiability of Inverse Function formula for the
function f (x) = x 3 +1
Solution:
y = x 3 +1
x = y3 +1
y = x −1
3
x = 3 y −1
f −1 (y) = 3 y −1
dy d  3 
=  x + 1 = 3x 2
dx dx
and
1
2
dx d  3
d
1
−
=  y − 1 =  ( y − 1) 3  = ( y − 1) 3
 dy 
 3
dy dy
(
)
2
2
dy
1
3 y−1 = 3 y−1 3 =
=3
( ) dx
dx
dy

9. MONOTONIC
FUNCTIONS:
Suppose that the domain of a function f is on an open interval I on which
f’(x) > 0 or on which f’(x) < 0. Then f is one-to-one, f-1(x) is differentiable
at all values of x in the range of f.
Example:
Consider the function
f (x) = x 5 + x +1
Solution:
f '(x) = 5x 4 +1
Since f’(x) > 0 on the entire domain, f(x) is
monotonic, therefore it has an inverse
.Show that f(x) is one-to=one function.

10. MONOTONIC
FUNCTIONS:
Suppose that the domain of a function f is on an open interval I on which
f’(x) > 0 or on which f’(x) < 0. Then f is one-to-one, f-1(x) is differentiable
at all values of x in the range of f.
Example:
Consider the function
f (x) = x 5 + x +1
Solution:
f '(x) = 5x 4 +1
Since f’(x) > 0 on the entire domain, f(x) is
monotonic, therefore it has an inverse
.Show that f(x) is one-to=one function.