Study Notes on Derivatives of Inverse Functions

2.6 Derivatives of Inverse Functions

2.6.1 Introduction

• Overview of Functions - Calculus investigates how output changes with input and the reversibility of functions, focusing on the natural exponential and logarithmic functions.

2.6.2 Basic Facts About Inverse Functions

• Definition of Functions - An inverse function $g$ for $f$ satisfies $g(f(a)) = a$ and $f(g(b)) = b$. Notation is $f^{-1}$. If $y = f(x)$, then $x = f^{-1}(y)$.

• Characteristics of Inverse Functions - A function has an inverse if it is one-to-one and onto. Their graphs are reflections across the line $y = x$. The natural exponential ($f(x) = e^x$) and natural logarithm ($f^{-1}(y) = \text{ln}(y)$) are inverse functions, meaning $y = e^x$ is equivalent to $x = \text{ln}(y)$ for y > 0.

2.6.3 Derivative of the Natural Logarithm Function

• Objective - The derivative of the natural logarithm function $g(x) = \text{ln}(x)$ is derived using the chain rule from $e^{g(x)} = x$. Differentiating both sides yields $e^{g(x)}g'(x) = 1$, which simplifies to $g'(x) = \frac{1}{e^{g(x)}} = \frac{1}{x}$ for positive $x$.

• Derivative Rule Note - For all x > 0, $\frac{d}{dx}[ \text{ln}(x)] = \frac{1}{x}$.

2.6.4 Inverse Trigonometric Functions and Their Derivatives

• Periodicity of Trigonometric Functions - Trigonometric functions require domain restriction to have inverses. For example, the inverse of $f(x) = \text{sin}(x)$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$ is the arcsine function, $f^{-1}(y) = \text{arcsin}(y)$.

• Derivative of the Arcsine Function - For all real $x$ such that -1 < x < 1, $\frac{d}{dx}[ \text{arcsin}(x)] = \frac{1}{\sqrt{1-x^2}}$.

2.6.5 The Link Between the Derivative of a Function and the Derivative of Its Inverse

• Reciprocal Slopes - If $f$ and $g$ are differentiable inverse functions, their slopes at corresponding points are reciprocal. The general rule states: if $y = f(x)$ and $x = g(y)$, then $g'(x) = \frac{1}{f'(g(x))}$.

2.6.6 Summary1

• Key derivatives and general rule:

  • For x > 0, $\frac{d}{dx}[ \text{ln}(x)] = \frac{1}{x}$ .

  • For -1 < x < 1, $\frac{d}{dx}[ \text{arcsin}(x)] = \frac{1}{\sqrt{1-x^2}}$.

  • For all real $x$, $\frac{d}{dx}[ \text{arctan}(x)] = \frac{1}{1+x^2}$.

  • If $g$ is the inverse of a differentiable function $f$, then $g'(x) = \frac{1}{f'(g(x))}$ for any $x$ in the domain of $g$.