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 cos⁡xx9\frac{\cos{x}}{{x}^{9}}?

Let's begin!



ddxcos⁡xx9\frac{d}{dx} \frac{\cos{x}}{{x}^{9}}

1
Use Quotient Rule to find the derivative of cos⁡xx9\frac{\cos{x}}{{x}^{9}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
x9(ddxcos⁡x)−cos⁡x(ddxx9)x18\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\cos{x} is −sin⁡x-\sin{x}.
−x9sin⁡x−cos⁡x(ddxx9)x18\frac{-{x}^{9}\sin{x}-\cos{x}(\frac{d}{dx} {x}^{9})}{{x}^{18}}

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
−x9sin⁡x−9x8cos⁡xx18\frac{-{x}^{9}\sin{x}-9{x}^{8}\cos{x}}{{x}^{18}}

Done
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