How to Derivate

This page provides a quick reference and general knowledge for differentiation (derivation) in calculus.

Basic Rules

Constant Rule

d/dx(c) = 0

Power Rule

d/dx(x^n) = n·x^(n-1)

Sum Rule

d/dx(f(x) + g(x)) = f'(x) + g'(x)

Difference Rule

d/dx(f(x) - g(x)) = f'(x) - g'(x)

Product Rule

d/dx(f(x)·g(x)) = f'(x)·g(x) + f(x)·g'(x)

Quotient Rule

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

Chain Rule

d/dx(f(g(x))) = f'(g(x))·g'(x)

Common Derivatives

d/dx(sin x) = cos x

d/dx(cos x) = -sin x

d/dx(tan x) = sec²x

d/dx(ln x) = 1/x

d/dx(e^x) = e^x

d/dx(a^x) = a^x · ln a

d/dx(arcsin x) = 1/√(1-x²)

d/dx(arccos x) = -1/√(1-x²)

d/dx(arctan x) = 1/(1+x²)

General Tips

  • • Break down complex expressions using the sum, product, quotient, and chain rules
  • • Remember to simplify your answer where possible
  • • Practice with a variety of functions to get comfortable with the rules

For more, see your calculus textbook or Wikipedia: Differentiation rules.