\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)


Ad blocker interference detected!

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.