Q106 / 160 · AP EAMCET 2021 (24 Aug Shift 2) · Mathematics

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

  • Correct answer
    A
    $(\tan x)^{\sin x}\{\sec x+(\cos x)(\log (\tan x))\}$
  • B
    $(\sin x)^{\tan x}\{\sec x+(\cos x)(\log (\tan x))\}$
  • C
    $(\tan x)^{\sin x}\{\sec x-(\cos x)(\log (\tan x))\}$
  • D
    $(\sin x)^{\tan x}\{\sec x-(\cos x)(\log (\tan x))\}$

Solution

We have, $y=(\tan x)^{\sin x} \Rightarrow y=e^{\sin x \log \tan x}$ $\frac{d y}{d x}=e^{\sin x \log \tan x}\left(\frac{\sin x}{\tan x} \sec ^2 x+\cos x \log \tan x\right)$ $\Rightarrow \frac{d y}{d x}=(\tan x)^{\sin x}(\sec x+\cos x \log \tan x)$

Hence, the correct answer is option: 'A'.

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