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Computing Derivatives

Computing Derivatives

At a Glance - Derivative of a Sum (or Difference) of Functions

Anytime we add two functions, we can find the derivative of the sum pretty easily, so long as we know the derivative of each function.

The derivative of a sum is just the sum of the derivatives: (f + g)' = f ' + g '.

It's just like the rule for taking the limits of sums. Derivatives are just a special kind of limit, so some (but not all) of the rules we used for evaluating limits can be used for derivatives as well.

When the function has more than two terms, and some weird combination of addition and subtraction, the process is similar. We find the derivatives of the individual terms, combine those derivatives by addition or subtraction as in the original function, and everything works out.

Example 1

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


Example 2

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


Example 3

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


Example 4

Find the derivative of the polynomial

f(x) = 6x7 + 5x4 – 3x2 + 5.


Exercise 1

Find the derivative of the function.

  • f(x) = 3x + 7

Exercise 2

Find the derivative of the function.

  •  f(x) = ex + ln x

Exercise 3

Find the derivative of the function.


Exercise 4

What's the derivative of the following function?

  • f(x) = x4 – 3x2

Exercise 5

Find the derivative of the function.


Exercise 6

Find the derivative of the function.

  •  f(x) = log2 x3log2 x9

Exercise 7

Find the derivative of the function.

  •  f(x) = logx – 2cos x

Exercise 8

Find the derivative of the function.

  • f(x) = 3cos x – 2x7 + 4x3

Exercise 9

Find the derivative of the function.


Exercise 10

Find the derivative of the function.

  • f(x) = x(x + 4) – (x – 1)2

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