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provide solution for RD Sharma maths class 12 chapter Differentiation exercise  10.2 question 30

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Answer: \operatorname{cosec} x

Hint: You must know the rules of solving derivation of logarithm and trigonometric function.

Given: \log (\operatorname{cosec} x-\cot x)

Solution:

Let  y=\log (\operatorname{cosec} x-\cot x)

Differentiate both sides,

\frac{d y}{d x}=\frac{d}{d x} \log (\operatorname{cosec} x-\cot x)

\frac{d y}{d x}=\frac{1}{(\operatorname{cosec} x-\cot x)} \frac{d}{d x}(\operatorname{cosec} x-\cot x)

\frac{d y}{d x}=\frac{1}{(\operatorname{cosec} x-\cot x)} \times\left(-\operatorname{cosec} x \cot x+\operatorname{cosec}^{2} x\right)

\begin{aligned} &\frac{d y}{d x}=\frac{\operatorname{cosec} x(\operatorname{cosec} x-\cot x)}{(\operatorname{cosec} x-\cot x)} \\ &\frac{d y}{d x}=\operatorname{cosec} x \end{aligned}

 

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