Question

what is $f(x)$ if $f(x)=\frac{40x^{5}}{e^{x}}$? select the correct answer below: $f(x)=40(\frac{(x^{5})(e^{x})-(e^{x})(5x^{4})}{e^{2x}})$ $f(x)=40(\frac{(5x^{4})-(x^{5})}{(e^{x})})$ $f(x)=\frac{405x^{4}}{e^{x}}$ $f(x)=\frac{40(e^{x})(5x^{4}) - 40(x)(x^{5})}{e^{x}}$

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 40x^{5}$, $u'=200x^{4}$, and $v = e^{x}$, $v'=e^{x}$.

Step2: Calculate derivative

$f'(x)=\frac{(200x^{4})e^{x}-(40x^{5})e^{x}}{(e^{x})^{2}}=40\frac{(5x^{4})-(x^{5})}{e^{x}}$.

Answer:

$f'(x)=40(\frac{(5x^{4})-(x^{5})}{e^{x}})$