# Problem of the Week

## Updated at Jun 9, 2014 10:55 AM

This week's problem comes from the calculus category.

How would you differentiate $$\frac{\cos{x}}{{x}^{9}}$$?

Let's begin!

$\frac{d}{dx} \frac{\cos{x}}{{x}^{9}}$

 1 Use Quotient Rule to find the derivative of $$\frac{\cos{x}}{{x}^{9}}$$. The quotient rule states that $$(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}$$.$\frac{{x}^{9}(\frac{d}{dx} \cos{x})-\cos{x}(\frac{d}{dx} {x}^{9})}{{x}^{18}}$2 Use Trigonometric Differentiation: the derivative of $$\cos{x}$$ is $$-\sin{x}$$.$\frac{-{x}^{9}\sin{x}-\cos{x}(\frac{d}{dx} {x}^{9})}{{x}^{18}}$3 Use Power Rule: $$\frac{d}{dx} {x}^{n}=n{x}^{n-1}$$.$\frac{-{x}^{9}\sin{x}-9{x}^{8}\cos{x}}{{x}^{18}}$Done(-x^9*sin(x)-9*x^8*cos(x))/x^18