Problem of the Week

Updated at May 16, 2016 9:47 AM

How can we solve for the derivative of x5ex\frac{{x}^{5}}{{e}^{x}}?

Below is the solution.



ddxx5ex\frac{d}{dx} \frac{{x}^{5}}{{e}^{x}}

1
Use Quotient Rule to find the derivative of x5ex\frac{{x}^{5}}{{e}^{x}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
ex(ddxx5)−x5(ddxex)e2x\frac{{e}^{x}(\frac{d}{dx} {x}^{5})-{x}^{5}(\frac{d}{dx} {e}^{x})}{{e}^{2x}}

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
5exx4−x5(ddxex)e2x\frac{5{e}^{x}{x}^{4}-{x}^{5}(\frac{d}{dx} {e}^{x})}{{e}^{2x}}

3
The derivative of ex{e}^{x} is ex{e}^{x}.
5exx4−x5exe2x\frac{5{e}^{x}{x}^{4}-{x}^{5}{e}^{x}}{{e}^{2x}}

Done
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