Question 1030346
I'll do parts A) and B) to get you started. I'll leave parts C) through E) for you to attempt on your own. If you get stuck, then you can post a separate question.



----------------------------------------------------------------------------------------------------------------------



Part A)



*[Tex \LARGE y = \ln|\csc(x) - \cot(x)|]



*[Tex \LARGE \frac{dy}{dx} = \frac{1}{\csc(x) - \cot(x)}*\frac{d}{dx}\left(\csc(x) - \cot(x)\right)]



*[Tex \LARGE \frac{dy}{dx} = \frac{1}{\csc(x) - \cot(x)}*\left(-\csc(x)*\cot(x) + \csc^2(x)\right)]



*[Tex \LARGE \frac{dy}{dx} = \frac{1}{\csc(x) - \cot(x)}*\csc(x)\left(\csc(x)-\cot(x)\right)]



*[Tex \LARGE \frac{dy}{dx} = \csc(x)*\frac{\csc(x)-\cot(x)}{\csc(x) - \cot(x)}]



*[Tex \LARGE \frac{dy}{dx} = \csc(x)]



----------------------------------------------------------------------------------------------------------------------



Part B)



*[Tex \LARGE y = \ln|\sin(x)|]



*[Tex \LARGE \frac{dy}{dx} = \frac{1}{\sin(x)}*\frac{d}{dx}\left(\sin(x)\right)]



*[Tex \LARGE \frac{dy}{dx} = \frac{1}{\sin(x)}*\cos(x)]



*[Tex \LARGE \frac{dy}{dx} = \frac{\cos(x)}{\sin(x)}]



*[Tex \LARGE \frac{dy}{dx} = \cot(x)]