Problem of the Week

Updated at Apr 11, 2016 8:15 AM

This week's problem comes from the calculus category.

How can we find the derivative of xcos⁡x\sqrt{x}\cos{x}?

Let's begin!



ddxxcos⁡x\frac{d}{dx} \sqrt{x}\cos{x}

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

2
Since x=x12\sqrt{x}={x}^{\frac{1}{2}}, using the Power Rule, ddxx12=12x−12\frac{d}{dx} {x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}
cos⁡x2x+x(ddxcos⁡x)\frac{\cos{x}}{2\sqrt{x}}+\sqrt{x}(\frac{d}{dx} \cos{x})

3
Use Trigonometric Differentiation: the derivative of cos⁡x\cos{x} is −sin⁡x-\sin{x}.
cos⁡x2x−xsin⁡x\frac{\cos{x}}{2\sqrt{x}}-\sqrt{x}\sin{x}

Done
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