Problem of the Week

Updated at Apr 27, 2015 9:06 AM

For this week we've brought you this calculus problem.

How can we find the derivative of ln⁡xsin⁡x\ln{x}\sin{x}?

Here are the steps:



ddxln⁡xsin⁡x\frac{d}{dx} \ln{x}\sin{x}

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

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

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

Done
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