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) = rac{1}{f'(f^{-1}(a))}

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

Why It Works

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

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

g'(x) = rac{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) = rac{1}{f'(1)} = rac{1}{3(1)^2 + 1} = rac{1}{4}

Inverse Function Derivatives 🎯

Key Takeaways — Part 1

1. $(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$
2. First find the $x$-value where $f(x) = a$
3. Then take the reciprocal of $f'$ at that point

Inverse Functions & Their Derivatives

Part 2 of 7 — Inverse Trigonometric Derivatives

Essential Formulas

rac{d}{dx}[arcsin x] = rac{1}{sqrt{1-x^2}}

rac{d}{dx}[arccos x] = -rac{1}{sqrt{1-x^2}}

rac{d}{dx}[arctan x] = rac{1}{1+x^2}

With Chain Rule

rac{d}{dx}[arctan(g(x))] = rac{g'(x)}{1+(g(x))^2}

Inverse Trig Derivatives 🎯

Key Takeaways — Part 2

1. Memorize the three inverse trig derivatives
2. Always apply chain rule with composite functions

Inverse Functions & Their Derivatives

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

Essential Derivatives

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

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

Logarithmic Differentiation

For complex products/quotients, take $\ln$ of both sides first.

Example: $y = \frac{x^2 \sqrt{x+1}}{(x-3)^4}$

$\ln y = 2\ln x + \frac{1}{2}\ln(x+1) - 4\ln(x-3)$

$\frac{y'}{y} = \frac{2}{x} + \frac{1}{2(x+1)} - \frac{4}{x-3}$

Exponential & Log Derivatives 🎯

Key Takeaways — Part 3

1. $e^x$ and $\ln x$ are inverses with simple derivatives
2. Always apply chain rule with composite functions

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.

To find $(f^{-1})'(7)$:

1. $f(2) = 7$, so $f^{-1}(7) = 2$
2. $(f^{-1})'(7) = \frac{1}{f'(2)} = \frac{1}{5}$

Table Problems 🎯

Using the table: $f(1) = 4$, $f'(1) = 3$, $f(2) = 7$, $f'(2) = 5$, $f(4) = 10$, $f'(4) = 2$.

Key Takeaways — Part 4

1. find the $x$ where $f(x) =$ the target value
2. Take reciprocal of $f'$ at that $x$

Inverse Functions & Their Derivatives

Part 5 of 7 — Integrals Leading to Inverse Trig

Key Antiderivatives

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

$\int \frac{1}{1+x^2}\,dx = \arctan x + C$

General Forms

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

$\int \frac{1}{a^2+x^2}\,dx = \frac{1}{a}\arctan\frac{x}{a} + C$

Inverse Trig Integrals 🎯

Key Takeaways — Part 5

1. Recognize when an integral leads to $\arcsin$ or $\arctan$
2. Use the general forms with parameter $a$

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! ✅