Problem of the Week

Updated at Jul 21, 2014 2:02 PM

How can we solve for the derivative of x8tan⁡x{x}^{8}\tan{x}?

Below is the solution.



ddxx8tan⁡x\frac{d}{dx} {x}^{8}\tan{x}

1
Use Product Rule to find the derivative of x8tan⁡x{x}^{8}\tan{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx8)tan⁡x+x8(ddxtan⁡x)(\frac{d}{dx} {x}^{8})\tan{x}+{x}^{8}(\frac{d}{dx} \tan{x})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
8x7tan⁡x+x8(ddxtan⁡x)8{x}^{7}\tan{x}+{x}^{8}(\frac{d}{dx} \tan{x})

3
Use Trigonometric Differentiation: the derivative of tan⁡x\tan{x} is sec⁡2x\sec^{2}x.
8x7tan⁡x+x8sec⁡2x8{x}^{7}\tan{x}+{x}^{8}\sec^{2}x

Done
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