Problem of the Week

Updated at May 1, 2017 2:03 PM

This week we have another calculus problem:

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

Let's start!



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

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

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

3
Since \(\sqrt{x}={x}^{\frac{1}{2}}\), using the Power Rule, \(\frac{d}{dx} {x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}\)
\[\frac{-\sqrt{x}\csc{x}\cot{x}-\frac{\csc{x}}{2\sqrt{x}}}{x}\]

Done
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