If y=( x)^ x , then d y d x is equal to - Vidyadip

MCQ

If $y=(\tan x)^{\sin x}$, then $\frac{d y}{d x}$ is equal to

A
$\sec x+\cos x$
B
$\sec x+\log \tan x$
C
$(\tan x)^{\sin x}$
✓
None of these

✓

Answer

Correct option: D.

None of these

(d) : We have, $y=(\tan x)^{\sin x}$
Taking $\log$ on both sides, we get
$\log y=\sin x \log (\tan x)$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{1}{y} \frac{d y}{d x} & =\frac{\sin x}{\tan x} \cdot \sec ^2 x+\cos x \log (\tan x) \\
& =(\tan x)^{\sin x}[\sec x+\cos x(\log \tan x)]
\end{aligned}
$

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