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
Cymath foriOSAndroid