# Problem of the Week

## Updated at Mar 13, 2017 3:23 PM

This week we have another calculus problem:

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

Let's start!

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

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