What is the derivative of f(x)+g(x) where f(x) and g(x) are differentiable?

Primary tabs

Lesson Parent: 
Differentiation IV: Differentiation Formulas Everyone Must Know
Problem: 

What is the derivative of f(x)+g(x) where f(x) and g(x) are differentiable?

Answer: 

f′(x)+g′(x)

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

Let's denote <Sign in to see all the formulas> by e(x) so that <Sign in to see all the formulas>. Our goal is to find the derivative of e(x) in terms of f′(x) and g′(x).

Recall that

<Sign in to see all the formulas>


Now let's plug in f(x) and g(x) to find that

<Sign in to see all the formulas>


The functions

<Sign in to see all the formulas>


have the following limits that exist

<Sign in to see all the formulas>


because we assume that f(x) and g(x) are differentiable. This is an important assumption needed in the fact that the function limit of the sum of two functions is the sum of their limits. We now use this fact to show that

<Sign in to see all the formulas>