# 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} $