Q95 / 180 · KCET 2022 · Mathematics
If $y=x^{\sin x}+(\sin x)^x$, then $\frac{d y}{d x}$ at $x=\frac{\pi}{2}$ is
- A$\frac{4}{\pi}$
- B$\pi \log \frac{\pi}{2}$
- Correct answerC1
- D$\frac{\pi^2}{2}$
Solution
Given, $y=x^{\sin x}+(\sin x)^x$
Let $u=x^{\sin x}$ and $v=(\sin x)^x$
Now, $u=x^{\sin x}, \log u=\sin x \log x$
Differentiating w.r.t. $x$, we get
$$
\begin{aligned}
& \frac{1}{u} \cdot \frac{d u}{d x}=\sin x \cdot \frac{1}{x}+\log x \times \cos x \\
& \frac{d u}{d x}=u\left(\frac{\sin x}{x}+\cos x \log x\right) \\
& \frac{d u}{d x}=x^{\sin x}\left(\frac{\sin x}{x}+\cos x \log x\right)
\end{aligned}
$$
Now, $v=(\sin x)^x \Rightarrow \log v=x \log (\sin x)$
Differentiating w.r.t. $x$, we get
$$
\frac{1}{v} \frac{d v}{d x}=x \frac{\cos x}{\sin x}+\log (\sin x)
$$
$\begin{aligned} & \frac{d v}{d x}=v(x \cot x+\log \sin x) \\ & \frac{d v}{d x}=(\sin x)^x(x \cot x+\log \sin x)\end{aligned}$
Adding Eqs. (i) and (ii), we get
$$
\begin{aligned}
& \frac{d u}{d x}+\frac{d v}{d x}=x^{\sin x}\left(\frac{\sin x}{x}+\right.+\cos x \log x) \\
&+(\sin x)^x(x \cot x+\log \sin x) \\
& \frac{d y}{d x}=x^{\sin x}\left(\frac{\sin x}{x}+\cos x \log x\right) \\
&++(\sin x)^x(x \cot x+\log \sin x)
\end{aligned}
$$
At $x=\frac{\pi}{2}$
$$
\begin{aligned}
\frac{d y}{d x} & =\left(\frac{\pi}{2}\right)\left(\frac{1}{\pi / 2}+0 \log \frac{\pi}{2}\right)+(1)^{\frac{\pi}{2}}\left(\frac{\pi}{2} \cdot 0+\log (1)\right) \\
\frac{d y}{d x} & =\left(\frac{\pi}{2}\right)\left(\frac{2}{\pi}+0\right)+1 \cdot(0+0)=1
\end{aligned}
$$
Hence, the correct answer is option: 'C'.