﻿ Differentiation Properties

Basic Differentiation Rules

The derivative of y = f(x) with respect to x can be represented several different ways:

Constant Function Rule

The derivative of a constant is zero. That is, if f(x) = C then f '(x) = 0.

See Derivation

Power Rule

If n is a real number, the derivative of xn is nxn-1. That is, if f(x) = xn then f '(x) = nxn-1.

See Derivation

Constant Multiple Rule

The derivative of a constant times a function is the constant times the derivative of the function. That is, if f(x) = kg(x) then f '(x) = kg'(x).

See Derivation

The derivative of the sum of two functions is the sum of the derivatives. That is,

if y = f(x) + g(x)

then y' = f '(x) + g'(x)

See Derivation

This property can be extended to the sum of two or more functions:

if f(x) = g1(x) + g2(x) + ... + gn(x)

then f '(x) = g1'(x) + g2'(x) + ... + gn'(x)

Since subtraction is mathematically equivalent to the addition of a negative, this rule generalizes to both addition and subtraction. This rule and the constant multiple rule means that if a function is a linear combination of other functions, the derivative of the function is the linear combination of the corresponding derivatives of the other functions:

Derivative of a Polynomial

We can use these four basic rules to find the derivative of a polynomial function: