# 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(ln(x)/(2*sqrt(x))-1/sqrt(x))/ln(x)^2