Math Problem Statement

What's the derivative of f^−1(x)

Solution

The derivative of the inverse of a function ff, denoted f1(x)f^{-1}(x), is given by the following formula:

ddx(f1(x))=1f(f1(x))\frac{d}{dx} \left( f^{-1}(x) \right) = \frac{1}{f'(f^{-1}(x))}

This formula applies under the assumption that ff is a one-to-one (bijective) function and that f(x)f'(x), the derivative of f(x)f(x), is non-zero at the relevant points. Here’s the step-by-step reasoning:

  1. Suppose y=f1(x)y = f^{-1}(x), which implies f(y)=xf(y) = x.
  2. Taking the derivative of both sides with respect to xx, we use implicit differentiation: ddx(f(y))=ddx(x)\frac{d}{dx}(f(y)) = \frac{d}{dx}(x)
  3. By the chain rule, this becomes: f(y)dydx=1f'(y) \cdot \frac{dy}{dx} = 1
  4. Solving for dydx\frac{dy}{dx}, we get: dydx=1f(y)=1f(f1(x))\frac{dy}{dx} = \frac{1}{f'(y)} = \frac{1}{f'(f^{-1}(x))}

This gives the derivative of the inverse function.

Would you like more details or have any questions?

Here are some related questions to further your understanding:

  1. How do you find the derivative of an inverse trig function?
  2. Can the formula for f1(x)f^{-1}(x) derivative be used for all functions?
  3. What happens if f(x)=0f'(x) = 0 at any point?
  4. How do you compute the derivative of an inverse function for a given example?
  5. How is implicit differentiation related to inverse functions?

Tip: Make sure that f(x)f(x) is one-to-one (bijective) before finding the derivative of its inverse!

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Derivatives
Implicit Differentiation

Formulas

Derivative of inverse function: d/dx(f^−1(x)) = 1 / f'(f^−1(x))

Theorems

Inverse Function Theorem

Suitable Grade Level

Undergraduate (Calculus I)