Question

What id the devirative of "TexFormula1" title="f(x) = e {}^{ - x} ln(x) " alt="f(x) = e {}^{ - x} ln(x) " align="absmiddle" class="latex-formula">

Answer 1

f'(x) = $e^{-x}$ / x - $e^{-x}$lnx

Differentiate using the product rule

given f(x) = g(x)h(x) then

f'(x) = g(x)h'(x) + h(x)g'(x) ← product rule

g(x) = $e^{-x}$ ⇒ g'(x) = - $e^{-x}$

h(x) = lnx ⇒ h'(x) = $\frac{1}{x}$

f'(x) = $e^{-x}$.$\frac{1}{x}$ - $e^{-x}$lnx

      = $e^{-x}$ / x - $e^{-x}$lnx