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 csc⁡xx5\frac{\csc{x}}{{x}^{5}}?

Let's start!



ddxcsc⁡xx5\frac{d}{dx} \frac{\csc{x}}{{x}^{5}}

1
Use Quotient Rule to find the derivative of csc⁡xx5\frac{\csc{x}}{{x}^{5}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
x5(ddxcsc⁡x)−csc⁡x(ddxx5)x10\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\csc{x} is −csc⁡xcot⁡x-\csc{x}\cot{x}.
−x5csc⁡xcot⁡x−csc⁡x(ddxx5)x10\frac{-{x}^{5}\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} {x}^{5})}{{x}^{10}}

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
−x5csc⁡xcot⁡x−5x4csc⁡xx10\frac{-{x}^{5}\csc{x}\cot{x}-5{x}^{4}\csc{x}}{{x}^{10}}

Done
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