A) \[{{(\sin x)}^{\tan x}}.(1+{{\sec }^{2}}x.\log \sin x)\]
B) \[\tan x.{{(\sin x)}^{\tan x-1}}.\cos x\]
C) \[{{(\sin x)}^{\tan x}}.{{\sec }^{2}}x.\log \sin x\]
D) \[\tan x.{{(\sin x)}^{\tan x-1}}\]
Correct Answer: A
Solution :
Given \[y={{(\sin x)}^{\tan x}}\]; \[\log y=\tan x.\log \sin x\] 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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