Question

i got problem with differential. i know that (e^x)'=e^x but why (e^4x)'=4e^x ?? whyn not 4e^x-1 like all the other equations? its just an exception?

Answer 1

\[\frac{d}{dx}e^{f(x)}=f'(x)\cdot e^{f(x)}\]
In this case, \(f(x)=4x\) and so \(f'(x)=4\), then
\[\frac{d}{dx}e^{4x}=4e^{4x},~\text{not }4e^x\]

Answer 2

It's a case of the derivative of a composition

\[(u\circ v)'= (u'\circ v)\cdot v'\\
%(u(v))'=u'(v)\cdot v'\]\[(e^{f})'\\
%=\exp(f(x))'\\
\qquad\qquad\qquad\qquad u(f)=e^f\qquad v=f\\
\qquad\qquad\qquad\qquad u'(f)=e^f\qquad v'=f'\\
(e^{f})'=e^{f}\cdot f'\]