Question

quotient rule: problem 1 (1 point) suppose that $f(x)=\frac{x + 5}{e^{x}}$. find $f(2)$. $f(2)=square$

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. For $f(x)=\frac{x + 5}{e^{x}}$, let $u=x + 5$ and $v = e^{x}$.

Step2: Find $u'$ and $v'$

Differentiate $u=x + 5$ with respect to $x$, we get $u'=1$. Differentiate $v = e^{x}$ with respect to $x$, we get $v'=e^{x}$.

Step3: Apply the quotient - rule

$f'(x)=\frac{u'v-uv'}{v^{2}}=\frac{1\times e^{x}-(x + 5)\times e^{x}}{(e^{x})^{2}}=\frac{e^{x}-(x + 5)e^{x}}{e^{2x}}=\frac{e^{x}(1-(x + 5))}{e^{2x}}=\frac{-x - 4}{e^{x}}$.

Step4: Evaluate $f'(2)$

Substitute $x = 2$ into $f'(x)$, we have $f'(2)=\frac{-2-4}{e^{2}}=-\frac{6}{e^{2}}$.

Answer:

$-\frac{6}{e^{2}}$