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 csc⁡xx\frac{\csc{x}}{\sqrt{x}}?

Let's start!



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

1
Use Quotient Rule to find the derivative of csc⁡xx\frac{\csc{x}}{\sqrt{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)x\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\csc{x} is −csc⁡xcot⁡x-\csc{x}\cot{x}.
−xcsc⁡xcot⁡x−csc⁡x(ddxx)x\frac{-\sqrt{x}\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} \sqrt{x})}{x}

3
Since x=x12\sqrt{x}={x}^{\frac{1}{2}}, using the Power Rule, ddxx12=12x−12\frac{d}{dx} {x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}
−xcsc⁡xcot⁡x−csc⁡x2xx\frac{-\sqrt{x}\csc{x}\cot{x}-\frac{\csc{x}}{2\sqrt{x}}}{x}

Done
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