Calculus · Quick Reference

Differentiation Cheat Sheet

Differential Calculus · Lesson 3/20

In one line: differentiation finds the derivative of a function — a measure of how fast the function's output changes as its input changes, i.e., its instantaneous rate of change or slope.

Key Ideas

1The Derivative as a Slope. f'(x) gives the slope of the tangent line to the graph of f at any point x.

2Definition via Limits. f'(x) = lim as h approaches 0 of [f(x+h) - f(x)] / h.

3Notation. Common notations include f'(x), dy/dx, and d/dx[f(x)] — they all mean the same thing.

4Differentiability. A function must be continuous to be differentiable, but continuity alone does not guarantee differentiability (sharp corners, for example).

5Power Rule Preview. For f(x) = x^n, f'(x) = n*x^(n-1). This is the single most-used shortcut in differentiation.

Worked Examples

Use the limit definition to find the derivative of f(x) = x^2.

f'(x) = 2x

Find the derivative of f(x) = x^3 using the power rule.

f'(x) = 3x^2

Find the slope of the tangent line to f(x) = x^2 at x = 4.

Slope = 8

Formulas

f'(x) = lim (h->0) [f(x+h) - f(x)] / h

d/dx [x^n] = n*x^(n-1)

Practice Yourself

Find the derivative of f(x) = x^4.

4x^3

Find the derivative of f(x) = x (i.e., x^1).

Find the derivative of a constant function f(x) = 7.