Problem of the Week

Updated at Oct 6, 2014 8:19 AM

To get more practice in calculus, we brought you this problem of the week:

How would you differentiate xln⁡x\frac{\sqrt{x}}{\ln{x}}?

Check out the solution below!



ddxxln⁡x\frac{d}{dx} \frac{\sqrt{x}}{\ln{x}}

1
Use Quotient Rule to find the derivative of xln⁡x\frac{\sqrt{x}}{\ln{x}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
ln⁡x(ddxx)−x(ddxln⁡x)ln⁡x2\frac{\ln{x}(\frac{d}{dx} \sqrt{x})-\sqrt{x}(\frac{d}{dx} \ln{x})}{{\ln{x}}^{2}}

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}}
ln⁡x2x−x(ddxln⁡x)ln⁡x2\frac{\frac{\ln{x}}{2\sqrt{x}}-\sqrt{x}(\frac{d}{dx} \ln{x})}{{\ln{x}}^{2}}

3
The derivative of ln⁡x\ln{x} is 1x\frac{1}{x}.
ln⁡x2x−1xln⁡x2\frac{\frac{\ln{x}}{2\sqrt{x}}-\frac{1}{\sqrt{x}}}{{\ln{x}}^{2}}

Done
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