Wolfram Alpha:

```Derivatives of Inverse Functions
--------------------------------

Property of inverses:  f(g(x)) = g(f(x)) = x

d[f(g(x))]/dx = d[x]/dx

Chain Rule:  f'(g(x)).g'(x) = 1

∴ g'(x) = 1/f'(g(x))

Example 1.

Find g'(x) given f(x) = x2 - 81

∴ g(x) = ±√(x + 81)

f'(g(x)) = f'(±√(x + 81)) = ±1/2√(x + 81)

g'(x) = 1/f'(g(x)) = ±2√(x + 81)

Example 2.

Find g'(x) given f(x) = x6

∴ g(x) = x1/6

f'(g(x)) = f'(x1/6) = x-5/6/6

g'(x) = 1/f'(g(x)) = 6/x-5/6 = 6x5/6

Example 3.

Find g'(2) given f(x) = x3 + x, g(x) = f-1(x), g(2) = 1

x = y3 + y

∴ dx = 3y2dy + dy

∴ dx = (3y2 + 1)dy

∴ dy/dx = 1/(3y2 + 1)

Swap y for g(x):

dg(x)/dx = 1/(3g(x)2dy + 1)

∴ g'(2) = 1/3(g(2)2dy + 1) = 1/4

Example 4.

Find g'(x) given f(x) = ex

∴ g(x) = lnx

f'(g(x)) = f'(lnx) = 1/x

g'(x) = 1/f'(g(x)) = x

Example 5.

Find g'(x) given f(x) = 2arcsin(x)

g(x) = sin(x/2)

f'(g(x)) = f'(sin(x/2)) = cos(x/2)/2

Example 6.

Find f'(x) given y = 2arcsin(x)

Here we use the inverse function to find the derivative.

x = sin(y/2)

dx = cos(y/2)dy/2

dy/dx = 2/cos(y/2)

= 2/cos(arcsin(x))

= 2/√(cos2(arcsin(x)))

= 2/√(1 - sin2(arcsin(x)))

= 2/√(1 - x2)```