## 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 x^{n} is nx^{n-1}. That is, if f(x) =
x^{n} then f '(x) = nx^{n-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

### Addition Rule

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) = g_{1}(x) + g_{2}(x) + ... + g_{n}(x)

then
f '(x) = g_{1}'(x) + g_{2}'(x) + ... + g_{n}'(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: