# The Derivative Sum Rule For functions $f$ and $g$ of $x$, the derivative of their sum equals the sum of their derivatives: $\frac d {dx} \bigg[f + g\bigg] = \frac {df} {dx} + \frac {dg}{dx}$ ^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 {df} {dx} + \frac {dg} {dx} \end{align} $