Question

homework4: problem 4
(1 point)
differentiate ( y=csc x(x+cot x) ).
( y^{prime}= ) ( x csc (x) cot (x)-csc (x) cot ^{2}(x)+csc (x)-csc ^{3}(x) )
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Explanation:

Step1: Apply the product rule

The product rule states that if \(y = u\cdot v\), then \(y'=u'v + uv'\). Let \(u = \csc x\) and \(v=x+\cot x\). First, find \(u'\) and \(v'\).
The derivative of \(\csc x\) is \(-\csc x\cot x\) (i.e., \(u'=-\csc x\cot x\)), and the derivative of \(v=x+\cot x\) is \(v' = 1-\csc^{2}x\) (since the derivative of \(x\) is \(1\) and the derivative of \(\cot x=-\csc^{2}x\)).

Step2: Substitute into the product rule formula

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Answer:

\(y'=-x\csc x\cot x-\csc x\cot^{2}x+\csc x-\csc^{3}x\)