Problem of the Week

Updated at Aug 4, 2014 8:03 AM

This week we have another calculus problem:

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

Let's start!



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

1
Use Constant Factor Rule: \(\frac{d}{dx} cf(x)=c(\frac{d}{dx} f(x))\).
\[\frac{1}{5}(\frac{d}{dx} \frac{\csc{x}}{x})\]

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

3
Use Trigonometric Differentiation: the derivative of \(\csc{x}\) is \(-\csc{x}\cot{x}\).
\[\frac{1}{5}\times \frac{-x\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} x)}{{x}^{2}}\]

4
Use Power Rule: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\).
\[\frac{-x\csc{x}\cot{x}-\csc{x}}{5{x}^{2}}\]

Done
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