# 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(5*e^x*x^4-x^5*e^x)/e^(2*x)