Derivada de una suma

[Yanes]

Sean
$f:\text{Dom}_1\subseteq \mathbb{R} \to \text{Im}_1\subseteq \mathbb{R}$
$f':\text{Dom}_2\subseteq \text{Dom}_1\to \text{Im}_2\subseteq \text{Im}_1~/~f'(x)=\lim \limits _{h\to 0}\frac{f(x+h)-f(x)}{h}$
$g(x)=f_1(x)+f_2(x)\Rightarrow g'(x)=f_1'(x)+f_2'(x)$

Demostración:

Spoiler: mostrar

$g(x)=f_1(x)+f_2(x)$
$g'(x)=\lim \limits _{h\to 0}\frac{(f_1(x+h)+f_2(x+h))-(f_1(x)+f_2(x))}{h}$
$g'(x)=\lim \limits _{h\to 0}\frac{f_1(x+h)+f_2(x+h)-f_1(x)-f_2(x)}{h}$
$g'(x)=\lim \limits _{h\to 0}\frac{f_1(x+h)-f_1(x)+f_2(x+h)-f_2(x)}{h}$
$g'(x)=\lim \limits _{h\to 0}\left (\frac{f_1(x+h)-f_1(x)}{h}+\frac{f_2(x+h)-f_2(x)}{h}\right )$

Sabemos que el límite de una suma es igual a la suma de límites https://www.omaforos.com.ar/viewtopic.php?f=38&t=6393

$g'(x)=\lim \limits _{h\to 0}\frac{f_1(x+h)-f_1(x)}{h}+\lim \limits _{h\to 0}\frac{f_2(x+h)-f_2(x)}{h}$
$g'(x)= f_1'(x)+f_2'(x)$