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 \(\frac{\sqrt{x}}{\ln{x}}\)?

Check out the solution below!



\[\frac{d}{dx} \frac{\sqrt{x}}{\ln{x}}\]

1
Use Quotient Rule to find the derivative of \(\frac{\sqrt{x}}{\ln{x}}\). The quotient rule states that \((\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}\).
\[\frac{\ln{x}(\frac{d}{dx} \sqrt{x})-\sqrt{x}(\frac{d}{dx} \ln{x})}{{\ln{x}}^{2}}\]

2
Since \(\sqrt{x}={x}^{\frac{1}{2}}\), using the Power Rule, \(\frac{d}{dx} {x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}\)
\[\frac{\frac{\ln{x}}{2\sqrt{x}}-\sqrt{x}(\frac{d}{dx} \ln{x})}{{\ln{x}}^{2}}\]

3
The derivative of \(\ln{x}\) is \(\frac{1}{x}\).
\[\frac{\frac{\ln{x}}{2\sqrt{x}}-\frac{1}{\sqrt{x}}}{{\ln{x}}^{2}}\]

Done
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