# Problem of the Week

## Updated at May 11, 2015 1:27 PM

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

Below is the solution.

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

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