Question: "Derivatives involving other trigonometric functions Find the derivative..." - Plainmath

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Answered question

2021-10-21

Derivatives involving other trigonometric functions Find the derivative of the following functions.

y = \frac{\cot x}{1 + \csc x}

Answer & Explanation

timbalemX

Skilled2021-10-22Added 108 answers

Step 1
Given:

y = \frac{\cot x}{1 + \csc x}

Step 2
Differentiate the function using quotient rule.

\frac{d y}{d x} = \frac{(1 + \csc x) \frac{d}{d x} (\cot x) - (\cot x) \frac{d}{d x} (1 + \csc x)}{{(1 + \csc x)}^{2}}

Simplify further:

y^{'} = \frac{{(- \csc x)}^{2} (1 + \csc x) - (- \cot x \csc x) (\cot x)}{{(1 + \csc x)}^{2}}

= \frac{- \csc^{2} (x) (1 + \csc x) + \cot^{2} x \csc x}{{(1 + \csc x)}^{2}}

Therefore,

y^{'} = \frac{- \csc^{2} (x) (1 + \csc x) + \cot^{2} x \csc x}{{(1 + \csc x)}^{2}}

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