How do you differentiate $f (x) = \frac{1}{x} \cdot 5 x \cdot e^{x}$ $f(x)=1/x5xe^x$ using the product rule?

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Answer 1 · 2016-08-28T10:31:47

$5 e^{x}$ $5e^x$

$\frac{d}{d x} (\frac{1}{x} 5 x e^{x})$ $frac{d}{dx}(frac{1}{x}5xe^x)$

taking the constant out, $(acdot f)^'=acdot f^'$

$=5frac{d}{dx}(frac{1}{x}xe^x)$

applying product rule, $(fcdot g)^'=f^'cdot g+fcdot g^'$

$=5(frac{d}{dx}(frac{1}{x})xe^x+frac{d}{dx}(xe^x)frac{1}{x})$

we know,
$frac{d}{dx}(frac{1}{x})=-frac{1}{x^2}$
and,
$frac{d}{dx}(xe^x)=e^x+e^x*x$

finally,
$=5((-frac{1}{x^2})xe^x+(e^x+e^x*x)frac{1}{x})$

solving it,
$5e^x$

How do you differentiate f(x)=1/x*5x*e^xf(x)=1x⋅5x⋅exf(x)=1/x*5x*e^x using the product rule?