Problem of the Week

Updated at Jun 29, 2015 9:15 AM

This week's problem comes from the calculus category.

How can we find the derivative of \(\frac{\csc{x}}{\cos{x}}\)?

Let's begin!



\[\frac{d}{dx} \frac{\csc{x}}{\cos{x}}\]

1
Use Quotient Rule to find the derivative of \(\frac{\csc{x}}{\cos{x}}\). The quotient rule states that \((\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}\).
\[\frac{\cos{x}(\frac{d}{dx} \csc{x})-\csc{x}(\frac{d}{dx} \cos{x})}{\cos^{2}x}\]

2
Use Trigonometric Differentiation: the derivative of \(\csc{x}\) is \(-\csc{x}\cot{x}\).
\[\frac{-\cos{x}\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} \cos{x})}{\cos^{2}x}\]

3
Use Trigonometric Differentiation: the derivative of \(\cos{x}\) is \(-\sin{x}\).
\[\frac{-\cos{x}\csc{x}\cot{x}+\csc{x}\sin{x}}{\cos^{2}x}\]

Done
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