Calc2.59

From Exampleproblems

$h(x)=f(x)+g(x)\,$

$f'(x)=\lim_{\Delta x \to 0}\frac{h(x+\Delta x)-h(x)}{\Delta x}=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)+g(x+\Delta x)-(f(x)+g(x))}{\Delta x}$

$=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}+\lim_{\Delta x \to 0}\frac{g(x+\Delta x)-g(x)}{\Delta x}=f'(x)+g'(x)$

Thus, the derivative of a sum is the sum of the derivatives.

$(f(x)+g(x))'=f'(x)+g'(x)\,$