Problem of the Week

Updated at May 23, 2016 12:02 PM

How can we solve for the derivative of ln⁡xcos⁡x\ln{x}\cos{x}?

Below is the solution.



ddxln⁡xcos⁡x\frac{d}{dx} \ln{x}\cos{x}

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

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

3
Use Trigonometric Differentiation: the derivative of cos⁡x\cos{x} is −sin⁡x-\sin{x}.
cos⁡xx−ln⁡xsin⁡x\frac{\cos{x}}{x}-\ln{x}\sin{x}

Done
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