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

✓

Answer

$y=\log \left(\sin e^x\right)$
Differentiating w.r.t. $x$, we get
$\begin{aligned}
\frac{d y}{d x}= & \frac{1}{\sin e^x} \cdot \frac{d}{d x}\left(\sin e^x\right)=\frac{1}{\sin e^x} \cos e^x \cdot \frac{d}{d x} e^x=\frac{1}{\sin e^x} \cos e^x \cdot e^x \\
& =e^x \cot e^x
\end{aligned}
$

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