Problem of the Week

Updated at Dec 19, 2016 3:21 PM

For this week we've brought you this calculus problem.

How can we find the derivative of csc⁡x−x3\csc{x}-{x}^{3}?

Here are the steps:



ddxcsc⁡x−x3\frac{d}{dx} \csc{x}-{x}^{3}

1
Use Sum Rule: ddxf(x)+g(x)=(ddxf(x))+(ddxg(x))\frac{d}{dx} f(x)+g(x)=(\frac{d}{dx} f(x))+(\frac{d}{dx} g(x)).
(ddxcsc⁡x)+(ddx−x3)(\frac{d}{dx} \csc{x})+(\frac{d}{dx} -{x}^{3})

2
Use Trigonometric Differentiation: the derivative of csc⁡x\csc{x} is −csc⁡xcot⁡x-\csc{x}\cot{x}.
−csc⁡xcot⁡x+(ddx−x3)-\csc{x}\cot{x}+(\frac{d}{dx} -{x}^{3})

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
−csc⁡xcot⁡x−3x2-\csc{x}\cot{x}-3{x}^{2}

Done
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