Problem of the Week

Updated at May 11, 2015 1:27 PM

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

Below is the solution.



ddxcsc⁡xtan⁡x\frac{d}{dx} \frac{\csc{x}}{\tan{x}}

1
Use Quotient Rule to find the derivative of csc⁡xtan⁡x\frac{\csc{x}}{\tan{x}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
tan⁡x(ddxcsc⁡x)−csc⁡x(ddxtan⁡x)tan⁡2x\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\csc{x} is −csc⁡xcot⁡x-\csc{x}\cot{x}.
−tan⁡xcsc⁡xcot⁡x−csc⁡x(ddxtan⁡x)tan⁡2x\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\tan{x} is sec⁡2x\sec^{2}x.
−tan⁡xcsc⁡xcot⁡x−csc⁡xsec⁡2xtan⁡2x\frac{-\tan{x}\csc{x}\cot{x}-\csc{x}\sec^{2}x}{\tan^{2}x}

Done
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