AP CALCULUS · MODULE 02A

Derivatives:
Concepts & Rules

Limit definition, tangent lines, differentiation rules, and techniques for finding derivatives.

📘 BUders Learning System

This module covers the definition of the derivative and all the basic differentiation rules. Study each topic in order for best results.

What is a Derivative?

Limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$, tangent lines, differentiability.

definition → start

Basic Differentiation Rules

Power rule: $\frac{d}{dx}x^n = nx^{n-1}$, constant multiple rule, sum/difference rules, derivatives of $e^x$ and $\ln x$.

rules → start

Product & Quotient Rules

$(fg)' = f'g + fg'$, $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.

rules → start

Derivatives of Trig Functions

$\frac{d}{dx}\sin x = \cos x$, $\frac{d}{dx}\cos x = -\sin x$, $\frac{d}{dx}\tan x = \sec^2 x$, plus sec, csc, cot.

trigonometry → start

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ — derivative of compositions (including trig examples).

composition → start

Implicit Differentiation

Differentiating equations where $y$ is not isolated (circles, ellipses, etc.).

implicit → start

Derivatives of Inverse Functions

$(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$, inverse trig derivatives: $\arcsin x$, $\arccos x$, $\arctan x$.

inverse functions → start

Practice Problems

20 solved problems covering all derivative rules and techniques.

practice → start

20-question interactive quiz with instant feedback.

assessment → start