Constant Multiple Rule

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The Constant Multiple Rule. If `c` is a constant anf `f` is a differentiable function then `(cf(x))'=c (f(x))'` .

Proof. By definition `(cf(x))'=lim_(h->0)(cf(x+h)-cf(x))/h=c lim_(h->0)(f(x+h)-f(x))/h=cf'(x)`.

Example 1. Find `f'(x)` if `f(x)=2*3^x`.

`f'(x)=(2*3^x)'=2(3^x)'=2*3^x ln(3)`.

Example 2. Find `f'(x)` if `f(x)=1/(2sqrt(x))`.

`f'(x)=(1/(2sqrt(x)))'=1/2(1/sqrt(x))'=1/2(1/(x^(1/2)))'=1/2(x^(-1/2))'=1/2*(-1/2)x^(-1/2-1)=-1/4 x^(-3/2)`.