# Sum and Difference rule for differentiable equations

The sum and difference rule for differentiable equations states:

The sum (or difference) of two differentiable functions is differentiable and [its derivative] is the sum (or difference) of their derivatives.

$$\frac{\text{d}}{\text{d}x}[f(x) + g(x)] = f'(x) + g'(x)$$ $$\frac{\text{d}}{\text{d}x}[f(x) - g(x)] = f'(x) - g'(x)$$

What is the proof of this rule?

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When I saw it, we just plugged f(x)+g(x) into the definition of derivative and out it came in a couple lines. –  Ross Millikan Jun 17 '11 at 15:43

So let $\nu(x)=f(x)+g(x)$. We have
Similarly you can do it for $f(x)-g(x)$ case. An Exercise is awaiting you, here:
Exercise. Prove that if $f$ and $g$ are differentiable, then their product $fg$ is also differentiable.