Problem of the Week

Updated at Jul 13, 2015 8:03 AM

This week we have another calculus problem:

How can we find the derivative of csc⁡xx\frac{\csc{x}}{x}?

Let's start!



ddxcsc⁡xx\frac{d}{dx} \frac{\csc{x}}{x}

1
Use Quotient Rule to find the derivative of csc⁡xx\frac{\csc{x}}{x}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
x(ddxcsc⁡x)−csc⁡x(ddxx)x2\frac{x(\frac{d}{dx} \csc{x})-\csc{x}(\frac{d}{dx} x)}{{x}^{2}}

2
Use Trigonometric Differentiation: the derivative of csc⁡x\csc{x} is −csc⁡xcot⁡x-\csc{x}\cot{x}.
−xcsc⁡xcot⁡x−csc⁡x(ddxx)x2\frac{-x\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} x)}{{x}^{2}}

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
−xcsc⁡xcot⁡x−csc⁡xx2\frac{-x\csc{x}\cot{x}-\csc{x}}{{x}^{2}}

Done
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