Problem of the Week

Updated at Nov 16, 2015 12:31 PM

For this week we've brought you this calculus problem.

How would you differentiate cos⁡xx\frac{\cos{x}}{x}?

Here are the steps:



ddxcos⁡xx\frac{d}{dx} \frac{\cos{x}}{x}

1
Use Quotient Rule to find the derivative of cos⁡xx\frac{\cos{x}}{x}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
x(ddxcos⁡x)−cos⁡x(ddxx)x2\frac{x(\frac{d}{dx} \cos{x})-\cos{x}(\frac{d}{dx} x)}{{x}^{2}}

2
Use Trigonometric Differentiation: the derivative of cos⁡x\cos{x} is −sin⁡x-\sin{x}.
−xsin⁡x−cos⁡x(ddxx)x2\frac{-x\sin{x}-\cos{x}(\frac{d}{dx} x)}{{x}^{2}}

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
−xsin⁡x−cos⁡xx2\frac{-x\sin{x}-\cos{x}}{{x}^{2}}

Done
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