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  • Differentiate the following w.r.t.: <img alt="e^{x}\log \sqrt{x}\cdot \csc x" src="https://entrancecorner.oncod

Differentiate the following w.r.t.x:

e^{x}\log \sqrt{x}\cdot \csc x

Option: 1

\frac{1}{2}e^{x} \left[-\log x \cdot \csc x \cot x+\frac{\csc x}{x}+\csc x \log x\right]


Option: 2

\frac{1}{2}e^{x} \left[-\log x \cdot \sec x \tan x+\frac{\sec x}{x}+\sec x \log x\right]


Option: 3

\frac{1}{2}e^{x} \left[-\log x \cdot \csc^{2} x +\frac{\cot x}{x}+\cot x \log x\right]


Option: 4

Cannot be determined


Answers (1)

best_answer

Let  y= y= e^{x}\log \sqrt{x}\cdot \csc x

Firstly simplify the given term,

y= y= e^{x}\log \sqrt{x}\cdot \csc x
=\frac{1}{2}\left[e^x \log x \cdot \csc x\right]

Then take derivative of the given term,

\frac{d y}{d x}=\frac{1}{2}\left[e^x \log x \cdot \frac{d}{d x}(\csc x)+e^x \cdot \csc x \cdot \frac{d}{d x}(\log x)+\log x \cdot \csc x \cdot \frac{d}{d x}\left(e^x\right)\right]
=\frac{1}{2}\left[e^x \log x \cdot(\csc x \cot x)+e^x \cdot \csc x \cdot \frac{1}{x}+\log x \cdot \csc x \cdot e^x\right]
=\frac{1}{2} e^x\left[\log x \cdot \left ( -\csc x \cot x \right )+\frac{\csc x}{x}+\csc x \log x\right]

\frac{1}{2}e^{x} \left[-\log x \cdot \csc x \cot x+\frac{\csc x}{x}+\csc x \log x\right]

 

Posted by

shivangi.shekhar

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