Sum rule of derivatives/Proof

Theorem. (Sum rule of derivatives) '

Prerequisites: The limit definition of the derivative, $f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$ Proof. $\frac{d}{dx}\bigl(f(x)+g(x)\bigr)=f'(x)+g'(x)$ From the limit definition of the derivative: $\begin{align} &f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\\ &g'(x)=\lim_{h\to0}\frac{g(x+h)-g(x)}{h} \end{align}$ $\begin{align} \frac{d}{dx}\bigl(f(x)+g(x)\bigr)&=\lim_{h\to0}\frac{f(x+h)+g(x+h)-f(x)-g(x)}{h}\\ &=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}+\lim_{h\to0}\frac{g(x+h)-g(x)}{h}\\ &=f'(x)+g'(x)\end{align}$