Derivative Rules

Prerequisites

Cartesian Coordinates Show

Gradient of a straight line Show

Equation of a straight line Show

Derivatives Show

Derivative Rules

Introduction

  • Finding derivatives manually can be difficult and time consuming.
  • Luckily, there are several simple rules that can be applied to make the job a lot easier.

Functions

Polynomials
NameFunctionDerivative
Constantf(x) = cf'(x) = 0
Straight linef(x) = mx + bf'(x) = m
Quadraticf(x) = x^2f'(x) = 2x
Polynomialf(x) = x^nf'(x) = nx^(n - 1)
Square rootf(x) = sqrt(x)f'(x) = 1/(2*sqrt(x))
Exponentials and logarithms
NameFunctionDerivative
Natural exponentialf(x) = e^xf'(x) = e^x
Exponentialf(x) = a^xf'(x) = ln(a)a^x
Natural logarithmf(x) = ln(x)f'(x) = 1/x
Logarithmf(x) = log_a(x)f'(x) = 1/(x ln(a)
Trigonometry
NameFunctionDerivative
Sinef(x) = sin(x)f'(x) = cos(x)
Cosinef(x) = cos(x)f'(x) = -sin(x)
Tangentf(x) = tan(x)f'(x) = sec^2(x)
Inverse sinef(x) = sin-1(x)f'(x) = 1/sqrt(1−x^2)
Inverse cosinef(x) = cos-1(x)f'(x) = -1/sqrt(1−x^2)
Inverse tangentf(x) = tan-1(x)f'(x) = 1/(1 + x^2)
Rules
NameFunctionDerivative
Multiplication by a constantcf(x)cf'(x)
Sum rulef(x) + g(x)f'(x) g'(x)
Difference rulef(x) - g(x)f'(x) - g'(x)
Quotient rulef(x) / g(x)(f’(x)g(x) − f(x)g’(x)) / g^2(x)
Reciprocal rule1/f(x)-f'(x)/f^2(x)
Chain rulef(g(x))f’(g(x))g’(x)