If y= ( e^x ), then d y d x is: - Vidyadip

MCQ

If $y=\log \left(\sin e^x\right)$, then $\frac{d y}{d x}$ is:

A
$\cot e^{x}$
✓
$e ^{ x } \cot e ^{ x }$
C
$\operatorname{cosec} e ^{ x }$
D
$e ^{ x } \operatorname{cosec} e ^{ x }$

✓

Answer

Correct option: B.

$e ^{ x } \cot e ^{ x }$

$e ^{ x } \cot e ^{ x }$
$y=\log \left(\sin e^{x}\right)$
$\frac{d y}{d x}=\frac{d}{d x} \log \left(\sin e^{x}\right)$
$=\frac{1}{\sin e^2} \frac{d}{d x} \sin e^{x}$
$=\frac{1}{\sin e^2} \cos e^x \frac{d}{d x} e^{x}$
$=\cot e^x\left(e^x\right)$
$=e^{x} \cot e^{x}$

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