Calculus · Quick Reference

Rules of Differentiation Cheat Sheet

Differential Calculus · Lesson 4/20

In one line: instead of using the limit definition every time, a handful of rules let us differentiate almost any function quickly.

Key Ideas

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

2Constant Multiple Rule. d/dx[c*f(x)] = c*f'(x) — constants pass through the derivative unchanged.

3Sum/Difference Rule. d/dx[f(x) +/- g(x)] = f'(x) +/- g'(x) — differentiate term by term.

4Product Rule. d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x).

5Quotient Rule. d/dx[f(x)/g(x)] = [f'(x)*g(x) - f(x)*g'(x)] / [g(x)]^2.

Differentiate f(x) = 5x^3 + 2x^2 - 7x + 4.

f'(x) = 15x^2 + 4x - 7

Differentiate f(x) = x^2 * sin(x) using the product rule.

f'(x) = 2x*sin(x) + x^2*cos(x)

Differentiate f(x) = x / (x+1) using the quotient rule.

f'(x) = 1 / (x+1)^2

d/dx[c*f(x)] = c*f'(x)

d/dx[f+g] = f' + g'

d/dx[f*g] = f'g + fg'

d/dx[f/g] = (f'g - fg') / g^2

Differentiate f(x) = 4x^3 - 2x + 1.

12x^2 - 2

Differentiate f(x) = x^2 * cos(x) using the product rule.

2x*cos(x) - x^2*sin(x)

Differentiate f(x) = (x+2)/(x-1) using the quotient rule.

-3 / (x-1)^2