Calc2.59

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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)\,


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