# The Derivative Sum Rule
The derivative of the sum of two functions equals the sum of their derivatives.
$\frac d {dx} \bigg[f(x) + g(x)\bigg] = f'(x) + g'(x)$ ^ke50ua
## Proof
As per the definition of the [[Derivative]]:
![[Derivative#^zzz13y]]
so
$
\begin{align}
\frac d {dx} \bigg[(f+g)(x)\bigg] &= \lim_{x \to a} \frac {(f+g)(x) - (f+g)(a)}{x-a}\\
&= \lim_{x \to a} \frac {f(x) + g(x) - (f(a) + g(a))}{x-a} \\
&= \lim_{x \to a} \frac {f(x) - f(a)}{x-a} + \lim_{x \to a} \frac {g(x) - g(a)}{x-a} \\
&= \frac d {dx} [f(x)] + \frac d {dx}[g(x)]
\end{align}
$