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Sum rule of derivatives/Proof

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Theorem. (Sum rule of derivatives) '
Prerequisites:
The limit definition of the derivative, f'(x)=\lim_{h\to0}\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}


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