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

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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

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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

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

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The functions

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have the following limits that exist

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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

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