The formula for the derivative of the inverse function, 𝑔(𝑥) g ( x ) , to a function 𝑓(𝑥) f ( x ) , is 𝑔′(𝑥)=1𝑓′(𝑔(𝑥)). g ′ ( x ) = 1 f ′ ( g ( x ) ).

Question

The formula for the derivative of the inverse function g(x) to a function f(x) is :

g’(x) = 1 / f’(g(x))

Use this formula to calculate g’(x) where g(x) is the inverse of f(x):

f(x) = 1 / x^2

g’(x) = ?

Answer 1

f(x) = x^-2 g(x) = x^-1/2

df/dx = -2/x³, so df/dg = -2/g³ = -2/(x^-1/2)³ = -2x^3/2

1/(df/dg) = -1/2 x^-3/2 which is correct if we just take dg/dx

Equation comes from the derivative wrt x of f(g(x)) = x df/dg dg/dx = 1 and dg/dx = 1/(df(g)/dg)