We find the derivative of each term of f, then add those derivatives together:

The same idea works for differences. The derivative of a difference is the difference of the derivatives: (f - g)' = f ' - g'.

Example 2

Let f(x) = sin x - cos x. Find f ' (x).

We take the difference of the derivatives of the terms:

The fun comes in when we start combining other rules with the addition/subtraction rule.

Example 3

Find the derivative of the polynomial f(x) = 5x^{3 }- 4x^{2}.

We find the derivative of each piece, then combine. Notice that we're also using the rule for multiplication-by-a-constant!

Example 4

Find the derivative of the polynomial

f(x) = 6x^{7} + 5x^{4 }- 3x^{2} + 5.

We need to find the derivative of each term, and then combine those derivatives, keeping the addition/subtraction as in the original function. For the sake of organization, find the derivative of each term first: