Study Notes on Derivatives of Inverse Functions

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

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}.

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}}.

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))}.

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.