> [!NOTE] Theorem (sum rule) > Suppose $f,g:I \to \mathbb{R}$ are defined on the open interval $I$ and are [[Fréchet Differentiation|differentiable]] at $c\in I$. Then $f+g$ is differentiable at $c$ and its [[Derivative of Real Function|derivative]] is given by$(f+g)'(c)=f'(c)+g'(c)$ **Proof**: We have, $\frac{f(x)+g(x)-f(c)-g(c)}{x-c} = \frac{f(x) - f(c)}{x-c} + \frac{g(x)-g(c)}{x-c}$By [[Sum Rule for Limits of Real Functions]], as $x\to c$ this expression approaches $f'(c)+g'(c).$