Problem of the Week

Updated at Oct 27, 2014 9:11 AM

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

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

Here are the steps:



ddxcot⁡xcos⁡x\frac{d}{dx} \cot{x}\cos{x}

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

2
Use Trigonometric Differentiation: the derivative of cot⁡x\cot{x} is −csc⁡2x-\csc^{2}x.
−csc⁡2xcos⁡x+cot⁡x(ddxcos⁡x)-\csc^{2}x\cos{x}+\cot{x}(\frac{d}{dx} \cos{x})

3
Use Trigonometric Differentiation: the derivative of cos⁡x\cos{x} is −sin⁡x-\sin{x}.
−csc⁡2xcos⁡x−cot⁡xsin⁡x-\csc^{2}x\cos{x}-\cot{x}\sin{x}

Done
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