Problem of the Week

Updated at May 16, 2016 9:47 AM

How can we solve for the derivative of \(\frac{{x}^{5}}{{e}^{x}}\)?

Below is the solution.



\[\frac{d}{dx} \frac{{x}^{5}}{{e}^{x}}\]

1
Use Quotient Rule to find the derivative of \(\frac{{x}^{5}}{{e}^{x}}\). The quotient rule states that \((\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}\).
\[\frac{{e}^{x}(\frac{d}{dx} {x}^{5})-{x}^{5}(\frac{d}{dx} {e}^{x})}{{e}^{2x}}\]

2
Use Power Rule: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\).
\[\frac{5{e}^{x}{x}^{4}-{x}^{5}(\frac{d}{dx} {e}^{x})}{{e}^{2x}}\]

3
The derivative of \({e}^{x}\) is \({e}^{x}\).
\[\frac{5{e}^{x}{x}^{4}-{x}^{5}{e}^{x}}{{e}^{2x}}\]

Done
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