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

MCQ

If $y = {(\sin x)^{\tan x}}$, then ${{dy} \over {dx}}$ is equal to

✓
${(\sin x)^{\tan x}}.(1 + {\sec ^2}x.\log \sin x)$
B
$\tan x.{(\sin x)^{\tan x - 1}}.\cos x$
C
${(\sin x)^{\tan x}}.{\sec ^2}x.\log \sin x$
D
$\tan x.{(\sin x)^{\tan x - 1}}$

✓

Answer

Correct option: A.

${(\sin x)^{\tan x}}.(1 + {\sec ^2}x.\log \sin x)$

a
(a) Given $y = {(\sin x)^{\tan x}}$ $\Rightarrow$ $\log y = \tan x.\log \sin x$

Differentiate with respect to $x,$

$\frac{1}{y}.\frac{{dy}}{{dx}} = \tan x.\cot x + \log \sin x.{\sec ^2}x$

$\frac{{dy}}{{dx}} = {(\sin x)^{\tan x}}[1 + \log \sin x.{\sec ^2}x]$.

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