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
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