Inverse Functions & Derivatives

Inverse Functions & Their Derivatives

Part 1 of 7 — Derivative of an Inverse Function

The Key Formula

If $f$ and $g$ are inverses ($g = f^{-1}$), then:

$(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$

Or equivalently: if $f(b) = a$, then $(f^{-1})'(a) = \frac{1}{f'(b)}$.

Why It Works

If $f(g(x)) = x$, differentiate both sides:

$f'(g(x)) cdot g'(x) = 1$

$g'(x) = \frac{1}{f'(g(x))}$

Worked Example

$f(x) = x^3 + x$. Find $(f^{-1})'(2)$.

We need $f(b) = 2$: $b^3 + b = 2 implies b = 1$.

$(f^{-1})'(2) = \frac{1}{f'(1)} = \frac{1}{3(1)^2 + 1} = \frac{1}{4}$

Inverse Function Derivatives 🎯

Key Takeaways — Part 1

1. $(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$

Inverse Functions & Their Derivatives

Part 2 of 7 — Inverse Trigonometric Derivatives

Essential Formulas

$\frac{d}{dx}[arcsin x] = \frac{1}{sqrt{1-x^2}}$

Inverse Functions & Their Derivatives

Part 3 of 7 — $e^x$ and $\ln x$ Review

Essential Derivatives

$\frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[\ln x] = \frac{1}{x}$

Inverse Functions & Their Derivatives

Part 4 of 7 — Table-Based Inverse Problems

The AP Pattern

Given a table of $f$ and $f'$ values, find $(f^{-1})'$ at a point.

Inverse Functions & Their Derivatives

Part 5 of 7 — Integrals Leading to Inverse Trig

Key Antiderivatives

$\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C$

Inverse Functions & Their Derivatives

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Inverse Functions — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Inverse Functions — Complete! ✅