# Problem of the Week

## Updated at May 1, 2017 2:03 PM

This week we have another calculus problem:

How can we find the derivative of $$\frac{\csc{x}}{\sqrt{x}}$$?

Let's start!

$\frac{d}{dx} \frac{\csc{x}}{\sqrt{x}}$

 1 Use Quotient Rule to find the derivative of $$\frac{\csc{x}}{\sqrt{x}}$$. The quotient rule states that $$(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}$$.$\frac{\sqrt{x}(\frac{d}{dx} \csc{x})-\csc{x}(\frac{d}{dx} \sqrt{x})}{x}$2 Use Trigonometric Differentiation: the derivative of $$\csc{x}$$ is $$-\csc{x}\cot{x}$$.$\frac{-\sqrt{x}\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} \sqrt{x})}{x}$3 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{-\sqrt{x}\csc{x}\cot{x}-\frac{\csc{x}}{2\sqrt{x}}}{x}$Done(-sqrt(x)*csc(x)*cot(x)-csc(x)/(2*sqrt(x)))/x