Search any question & find its solution
Question:
Answered & Verified by Expert
If $y=(\sin x)^{\cos x}$, then $\frac{d y}{d x}$ is equal to.
Solution:
2672 Upvotes
Verified Answer
$$
\begin{aligned}
&\text \left.\begin{array}{rl}
& y=(\sin x)^{\cos x} \\
\Rightarrow & \log y=\log (\sin x)^{\cos x}=\cos x \log \sin x \\
\therefore & \frac{d}{d y} \log y \cdot \frac{d y}{d x}=\frac{d}{d x}(\cos x \cdot \log \sin x) \\
\Rightarrow & \frac{1}{y} \cdot \frac{d y}{d x}=\cos x \cdot \frac{d}{d x} \log \sin x+\log \sin x \cdot \frac{d}{d x} \cos x \\
= & \cos x \cdot \frac{1}{\sin x} \cdot \frac{d}{d x} \sin x+\log \sin x \cdot(-\sin x) \\
= & \cot x \cdot \cos x-\log (\sin x) \cdot \sin x \\
\therefore \quad \frac{d y}{d x}=y\left[\frac{\cos ^2 x}{\sin x}-\sin x \cdot \log (\sin x)\right] \\
= & (\sin x)^{\cos x}\left[\frac{\cos { }^2 x}{\sin x}\right]
\end{array}\right]
\end{aligned}
$$
\begin{aligned}
&\text \left.\begin{array}{rl}
& y=(\sin x)^{\cos x} \\
\Rightarrow & \log y=\log (\sin x)^{\cos x}=\cos x \log \sin x \\
\therefore & \frac{d}{d y} \log y \cdot \frac{d y}{d x}=\frac{d}{d x}(\cos x \cdot \log \sin x) \\
\Rightarrow & \frac{1}{y} \cdot \frac{d y}{d x}=\cos x \cdot \frac{d}{d x} \log \sin x+\log \sin x \cdot \frac{d}{d x} \cos x \\
= & \cos x \cdot \frac{1}{\sin x} \cdot \frac{d}{d x} \sin x+\log \sin x \cdot(-\sin x) \\
= & \cot x \cdot \cos x-\log (\sin x) \cdot \sin x \\
\therefore \quad \frac{d y}{d x}=y\left[\frac{\cos ^2 x}{\sin x}-\sin x \cdot \log (\sin x)\right] \\
= & (\sin x)^{\cos x}\left[\frac{\cos { }^2 x}{\sin x}\right]
\end{array}\right]
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.