# The Derivative Sum Rule
For functions $u$ and $v$ of $x$, the derivative of their sum equals the sum of their derivatives:
$\frac d {dx} \bigg[u + v\bigg] = du + dv$ ^ke50ua
## Proof
As per the definition of the [[Derivative]]:
![[Derivative#^zzz13y]]
so
$
\begin{align}
\frac d {dx} \bigg[f(x) + g(x)\bigg]
&= \lim_{h \to 0} \frac {f(x+h) + g(x+h) - (f(x) + g(x))} h\\
&= \lim_{h \to 0} \frac {f(x+h) - f(x) + g(x+h) - g(x)} h \\
&= \lim_{h \to 0} \frac {f(x+h) - f(x)}h +
\lim_{h \to 0} \frac {g(x+h) - g(x)}h \\
&= \frac {df} {dx} + \frac {dg} {dx}
\end{align}
$