If f(x) = x^-1/3, What is the Derivative of the Inverse of f(x)?

We will be using the concept of inverse function to solve this.

Answer: If f(x) = x^-1/3, then the Derivative of the Inverse of f(x) is −3x^-4.

Let's solve this step by step to find the answer.

Explanation: 

Given that, f(x) = x^-1/3 

Let y = f(x) = x^-1/3 

y = x^-1/3 

Cubing on both sides:

y³ = (x^-1/3)³ 

y³ = x^-1

x = 1/y³ 

x = y^-3  

We know from the definition of inverse function that: f(x) = y ⇒ x = f^-1(y)

f^-1(y) = y^-3 

Now change the variable from y → x,

∴ f^-1(x) = x^-3 

d/dx (f^-1(x)) = d/dx (x^-3)

= −3 x^{-3 -1}  

= −3 x^-4 

Thus, if f(x) = x^-1/3, then the derivative of the inverse of f(x) is −3x^-4.