Derivative of a Sum (or Difference) of Functions Examples

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

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 '.

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.

Find the derivative of the polynomial f(x) = 5x³ – 4x².

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

Find the derivative of the polynomial

f(x) = 6x⁷ + 5x⁴ – 3x² + 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:

(6x⁷)' = 42x⁶
(5x⁴)' = 20x³
(3x²)' = 6x
5' = 0 

Therefore