Problem of the Week

Updated at Aug 12, 2013 10:11 AM

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

How can we solve for the derivative of x7cos⁡x{x}^{7}\cos{x}?

Here are the steps:



ddxx7cos⁡x\frac{d}{dx} {x}^{7}\cos{x}

1
Use Product Rule to find the derivative of x7cos⁡x{x}^{7}\cos{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx7)cos⁡x+x7(ddxcos⁡x)(\frac{d}{dx} {x}^{7})\cos{x}+{x}^{7}(\frac{d}{dx} \cos{x})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
7x6cos⁡x+x7(ddxcos⁡x)7{x}^{6}\cos{x}+{x}^{7}(\frac{d}{dx} \cos{x})

3
Use Trigonometric Differentiation: the derivative of cos⁡x\cos{x} is −sin⁡x-\sin{x}.
7x6cos⁡x−x7sin⁡x7{x}^{6}\cos{x}-{x}^{7}\sin{x}

Done
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