Q104 / 160 · TS EAMCET 2024 (09 May Shift 2) · Mathematics

If $y(\cos x)^{\sin x}=(\sin x)^{\sin x}$ then the value of $\frac{d y}{d x}$ at $x=\frac{\pi}{4}$ is

  • A
    0
  • B
    1
  • Correct answer
    C
    $\sqrt{2}$
  • D
    $\frac{\sqrt{3}}{2}$

Solution

$y=(\tan x)^{\sin x}$ $\begin{aligned} & \log y=\sin x \log (\tan x) \\ & \frac{1}{y} y^{\prime}=\cos x \log (\tan x)+\frac{\sin x}{\tan x} \sec ^2 x \\ & y^{\prime}=(\tan x)^{\sin x}\{\cos x \log (\tan x)+\sec x\} \\ & \left.\Rightarrow \frac{d y}{d x}\right|_{x=\frac{\pi}{4}}=1 \\ & \left(\frac{1}{\sqrt{2}}\right)\left\{\frac{1}{\sqrt{2}} \log 1+\sqrt{2}\right\}=\sqrt{2}\end{aligned}$

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

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