Question

Find d / dx ( e^in(x)).

Answer 1

let's first off apply a log rule of cancellation, keeping in mind that, first off is ln(), not in(), and that ln() is just a shortcut to logₑ.

$\bf \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad \stackrel{\stackrel{\textit{we'll use this one}}{\downarrow }}{a^{log_a x}=x} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ e^{ln(x)}\implies e^{log_e(x)}\implies x \\\\\\ \cfrac{d}{dx}\left[ e^{ln(x)} \right]\implies \cfrac{d}{dx}[x]\implies 1$