# 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:

y^{3} = (x^{-1/3})^{3}

y^{3} = x^{-1}

x = 1/y^{3}

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}.

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