Problem of the Week

Updated at Aug 5, 2013 5:46 PM

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

How would you differentiate cos⁡xsin⁡x\cos{x}\sin{x}?

Here are the steps:



ddxcos⁡xsin⁡x\frac{d}{dx} \cos{x}\sin{x}

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

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

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

Done
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