Problem of the Week

Updated at Jun 8, 2015 4:03 PM

This week we have another calculus problem:

How can we find the derivative of \(\frac{\csc{x}}{{x}^{5}}\)?

Let's start!



\[\frac{d}{dx} \frac{\csc{x}}{{x}^{5}}\]

1
Use Quotient Rule to find the derivative of \(\frac{\csc{x}}{{x}^{5}}\). The quotient rule states that \((\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}\).
\[\frac{{x}^{5}(\frac{d}{dx} \csc{x})-\csc{x}(\frac{d}{dx} {x}^{5})}{{x}^{10}}\]

2
Use Trigonometric Differentiation: the derivative of \(\csc{x}\) is \(-\csc{x}\cot{x}\).
\[\frac{-{x}^{5}\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} {x}^{5})}{{x}^{10}}\]

3
Use Power Rule: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\).
\[\frac{-{x}^{5}\csc{x}\cot{x}-5{x}^{4}\csc{x}}{{x}^{10}}\]

Done
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