# Problem of the Week

## Updated at Jun 23, 2014 1:03 PM

This week we have another calculus problem:

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

Let's start!

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

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