Problem of the Week

Updated at Jan 8, 2018 4:38 PM

This week we have another calculus problem:

How can we solve for the derivative of x4tan⁡x{x}^{4}\tan{x}?

Let's start!



ddxx4tan⁡x\frac{d}{dx} {x}^{4}\tan{x}

1
Use Product Rule to find the derivative of x4tan⁡x{x}^{4}\tan{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx4)tan⁡x+x4(ddxtan⁡x)(\frac{d}{dx} {x}^{4})\tan{x}+{x}^{4}(\frac{d}{dx} \tan{x})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
4x3tan⁡x+x4(ddxtan⁡x)4{x}^{3}\tan{x}+{x}^{4}(\frac{d}{dx} \tan{x})

3
Use Trigonometric Differentiation: the derivative of tan⁡x\tan{x} is sec⁡2x\sec^{2}x.
4x3tan⁡x+x4sec⁡2x4{x}^{3}\tan{x}+{x}^{4}\sec^{2}x

Done
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