Problem of the Week

Updated at Jul 31, 2017 11:10 AM

This week's problem comes from the calculus category.

How would you differentiate 8xtan⁡x8x\tan{x}?

Let's begin!



ddx8xtan⁡x\frac{d}{dx} 8x\tan{x}

1
Use Constant Factor Rule: ddxcf(x)=c(ddxf(x))\frac{d}{dx} cf(x)=c(\frac{d}{dx} f(x)).
8(ddxxtan⁡x)8(\frac{d}{dx} x\tan{x})

2
Use Product Rule to find the derivative of xtan⁡xx\tan{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
8((ddxx)tan⁡x+x(ddxtan⁡x))8((\frac{d}{dx} x)\tan{x}+x(\frac{d}{dx} \tan{x}))

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
8(tan⁡x+x(ddxtan⁡x))8(\tan{x}+x(\frac{d}{dx} \tan{x}))

4
Use Trigonometric Differentiation: the derivative of tan⁡x\tan{x} is sec⁡2x\sec^{2}x.
8(tan⁡x+xsec⁡2x)8(\tan{x}+x\sec^{2}x)

Done
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