Question

let $f(x)=\frac{1}{x}e^{x}$. find $f(x)$. choose 1 answer: (a) $\frac{1}{x}e^{x}$ (b) $e^{x}(\frac{1}{x}-\frac{1}{x^{2}})$ (c) $-\frac{1}{x^{2}}e^{x}$ (d) $-\frac{1}{x^{2}}(\frac{1}{x}+e^{x})$

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = e^{x}$, $u'=e^{x}$, $v = x$, and $v' = 1$.

Step2: Calculate the derivative

$f'(x)=\frac{e^{x}\cdot x - e^{x}\cdot1}{x^{2}}=\frac{e^{x}(x - 1)}{x^{2}}=e^{x}(\frac{1}{x}-\frac{1}{x^{2}})$

Answer:

B. $e^{x}(\frac{1}{x}-\frac{1}{x^{2}})$