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  • Find the derivative of <img alt="cosec^{2}x" src="https://entrancecorner.oncodecogs.com/gif.latex?cosec

Find the derivative of cosec^{2}x  with respect to e^{secx}

Option: 1

-\frac{2}{e^{\cos x}} \cdot \sec x \cdot \operatorname{cosec}^2 x


Option: 2

-\frac{2}{e^{\sec x}} \cdot \cos x \cdot \operatorname{cosec}^2 x


Option: 3

-\frac{2}{e^{\sec x}} \cdot \cos ^3 x \cdot \operatorname{cosec}^4 x


Option: 4

 None of these


Answers (1)

best_answer

\begin{aligned} & \text { Let } u=\operatorname{cosec}^2 x, v=e^{\sec x} \\ & \frac{d u}{d x}=-2 \cot x \cdot \operatorname{cosec}^2 x, \frac{d v}{d x}=e^{\sec x} \cdot \sec x \tan x \\ & \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}} \\ & \frac{d u}{d v}=-\frac{2 \cot x \cdot \operatorname{cosec}^2 x}{e^{\sec x} \cdot \sec x \tan x} \\ & \frac{d u}{d v}=-\frac{2}{e^{\sec x}} \cdot \frac{\cos x}{\sin x} \cdot \frac{1}{\sin ^2 x} \cdot \frac{\cos ^2 x}{\sin x} \\ & \frac{d u}{d v}=-\frac{2}{e^{\sec x}} \cdot \frac{\cos ^3 x}{\sin ^4 x} \\ & \frac{d u}{d v}=-\frac{2}{e^{\sec x}} \cdot \cos ^3 x \cdot \operatorname{cosec}^4 x \\ & \end{aligned}

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rishi.raj

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