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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
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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