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Constant multiple rule of derivatives/Proof

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\frac{d}{dx}a\,f(x)=a\,f'(x), for every constant a.


Limit definition of the derivative, f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}


Let g(x)=a\,f(x) for some constant a. By the limit definition of the derivative:

f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}
a\,f'(x)=a\,\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h \to 0}\frac{a\,f(x+h)-a\,f(x)}{h}

To prove the proposition, it suffices to show that g'(x)=a\,f'(x).

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


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