How To Find The Derivative Of An Inverse Function

Finding the derivative of an inverse functionis a powerful technique that appears repeatedly in calculus, physics, and engineering. This article explains how to find the derivative of an inverse function step by step, provides the underlying formula, and walks through a concrete example. By the end, you will be able to compute the derivative of any invertible function confidently and understand why the method works.

Introduction

When a function (f) is one‑to‑one and continuous on an interval, it possesses an inverse function (f^{-1}) that “undoes” the action of (f). The derivative of this inverse, (\dfrac{d}{dx}f^{-1}(x)), tells us how quickly the input to (f) must change to produce a given change in the output. Knowing how to find the derivative of an inverse function not only simplifies many differentiation problems but also deepens intuition about the geometry of curves and their reflections across the line (y=x).

The Concept of Inverse Functions

An inverse function (f^{-1}) satisfies [ f\big(f^{-1}(x)\big)=x \quad\text{and}\quad f^{-1}\big(f(x)\big)=x ]
for every (x) in the appropriate domains. Graphically, the graph of (f^{-1}) is the reflection of the graph of (f) across the line (y=x). This symmetry plays a central role in deriving the formula for the derivative of the inverse.

Derivative of an Inverse Function – The Core Formula

The key relationship is expressed by the following theorem:

[ \boxed{\displaystyle \frac{d}{dx}f^{-1}(x)=\frac{1}{,f'\big(f^{-1}(x)\big)}} ]

provided that (f') is non‑zero at the point (f^{-1}(x)). In words, the derivative of the inverse at a point equals the reciprocal of the derivative of the original function evaluated at the corresponding inverse point. This formula is the cornerstone of how to find the derivative of an inverse function.

Why the Formula Holds

Starting from the identity (f\big(f^{-1}(x)\big)=x), differentiate both sides with respect to (x) using the chain rule:

[ f'\big(f^{-1}(x)\big)\cdot \frac{d}{dx}f^{-1}(x)=1. ]

Solving for (\frac{d}{dx}f^{-1}(x)) yields the reciprocal relationship shown above. The derivation assumes that (f') does not vanish at the relevant point, ensuring the inverse function is locally differentiable.

Step‑by‑Step Procedure

To apply the formula in practice, follow these systematic steps:

1. Verify invertibility – Confirm that (f) is one‑to‑one on the interval of interest and that (f') is continuous and non‑zero there.
2. Find the inverse function – Solve the equation (y=f(x)) for (x) in terms of (y); the resulting expression is (f^{-1}(y)). 3. Differentiate the original function – Compute (f'(x)).
3. Substitute the inverse point – Replace (x) in (f'(x)) with (f^{-1}(x)) (or the appropriate variable).
4. Take the reciprocal – The derivative of the inverse is the reciprocal of the expression obtained in step 4.

This procedure encapsulates how to find the derivative of an inverse function in a clear, repeatable manner.

Worked Example

Consider the function (f(x)=x^{3}+2x). We want the derivative of its inverse at (x=3).

1. Invertibility check – (f'(x)=3x^{2}+2) is always positive, so (f) is strictly increasing and invertible on (\mathbb{R}). 2. Find the inverse implicitly – Solve (y=x^{3}+2x) for (x). While an explicit algebraic expression is messy, we only need the value (x) that satisfies (f(x)=3). - Test (x=1): (1^{3}+2\cdot1=3). Hence (f(1)=3) and therefore (f^{-1}(3)=1). 3. Differentiate (f) – (f'(x)=3x^{2}+2).
2. Evaluate at the inverse point – (f'\big(f^{-1}(3)\big)=f'(1)=3(1)^{2}+2=5).
3. Reciprocal – (\displaystyle \frac{d}{dx}f^{-1}(3)=\frac{1}{5}=0.2).

Thus, the slope of the inverse curve at the point corresponding to (x=3) is 0.2. This example illustrates each step of how to find the derivative of an inverse function without explicitly solving for the inverse formula.

Common Pitfalls - Skipping the invertibility check – If (f') changes sign or vanishes, the inverse may not be differentiable, and the formula fails.

• Confusing variables – Remember that the derivative of the inverse is taken with respect to the output variable of the original function, not the input. - Misapplying the reciprocal – The reciprocal must be taken after substituting the inverse point; doing it prematurely leads to incorrect results.

Being aware of these traps ensures a smooth application of the method.

Frequently Asked Questions

Q1: Can the formula be used when (f) is defined piecewise?
A: Yes, provided each piece is invertible on its domain and the overall function remains one‑to‑one. You must treat each piece separately and verify continuity of the derivative at the boundaries.

**Q2: What if (f'(f^{-1}(x))=