# Problem of the Week

## Updated at Jun 8, 2015 4:03 PM

This week we have another calculus problem:

How can we find the derivative of $$\frac{\csc{x}}{{x}^{5}}$$?

Let's start!

$\frac{d}{dx} \frac{\csc{x}}{{x}^{5}}$

 1 Use Quotient Rule to find the derivative of $$\frac{\csc{x}}{{x}^{5}}$$. The quotient rule states that $$(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}$$.$\frac{{x}^{5}(\frac{d}{dx} \csc{x})-\csc{x}(\frac{d}{dx} {x}^{5})}{{x}^{10}}$2 Use Trigonometric Differentiation: the derivative of $$\csc{x}$$ is $$-\csc{x}\cot{x}$$.$\frac{-{x}^{5}\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} {x}^{5})}{{x}^{10}}$3 Use Power Rule: $$\frac{d}{dx} {x}^{n}=n{x}^{n-1}$$.$\frac{-{x}^{5}\csc{x}\cot{x}-5{x}^{4}\csc{x}}{{x}^{10}}$Done(-x^5*csc(x)*cot(x)-5*x^4*csc(x))/x^10