Problem of the Week

Updated at Jul 28, 2014 8:38 AM

This week we have another calculus problem:

How can we solve for the derivative of ln⁡xtan⁡x\ln{x}\tan{x}?

Let's start!



ddxln⁡xtan⁡x\frac{d}{dx} \ln{x}\tan{x}

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

2
The derivative of ln⁡x\ln{x} is 1x\frac{1}{x}.
tan⁡xx+ln⁡x(ddxtan⁡x)\frac{\tan{x}}{x}+\ln{x}(\frac{d}{dx} \tan{x})

3
Use Trigonometric Differentiation: the derivative of tan⁡x\tan{x} is sec⁡2x\sec^{2}x.
tan⁡xx+ln⁡xsec⁡2x\frac{\tan{x}}{x}+\ln{x}\sec^{2}x

Done
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