The derivative of y = f(x) with respect to x can be represented several different ways:
The derivative of a constant is zero. That is, if f(x) = C then f '(x) = 0.
If n is a real number, the derivative of xn is nxn-1. That is, if f(x) = xn then f '(x) = nxn-1.
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).
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)
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:
We can use these four basic rules to find the derivative of a polynomial function:
