Q.If $y = x^{\sin x} + (\sin x)^{\cos x}$, find $\dfrac{dy}{dx}$. OR Show that of all the rectangles inscribed in a given fixed circle the square has the maximum area.

Use logarithmic differentiation on each term separately (since both base and exponent are functions of $x$), then add the results. OR: an inscribed rectangle's area is maximised exactly when it becomes a square.

Given: $y=x^{\sin x}+(\sin x)^{\cos x}$. Find $\dfrac{dy}{dx}$.

Let $u=x^{\sin x}$ and $v=(\sin x)^{\cos x}$, so $y=u+v$.

For $u=x^{\sin x}$: take $\ln u=\sin x\ln x$. Differentiate both sides w.r.t. $x$:

$\dfrac{1}{u}\dfrac{du}{dx}=\cos x\ln x+\sin x\cdot\dfrac{1}{x}$

$\dfrac{du}{dx}=x^{\sin x}\left(\cos x\ln x+\dfrac{\sin x}{x}\right)$

For $v=(\sin x)^{\cos x}$: take $\ln v=\cos x\ln(\sin x)$. Differentiate:

$\dfrac{1}{v}\dfrac{dv}{dx}=-\sin x\ln(\sin x)+\cos x\cdot\dfrac{\cos x}{\sin x}$

$\dfrac{dv}{dx}=(\sin x)^{\cos x}\left(\dfrac{\cos^2x}{\sin x}-\sin x\ln(\sin x)\right)$

Adding:

$\dfrac{dy}{dx}=x^{\sin x}\left(\cos x\ln x+\dfrac{\sin x}{x}\right)+(\sin x)^{\cos x}\left(\dfrac{\cos^2x}{\sin x}-\sin x\ln(\sin x)\right)$

OR (alternative version of this question): Show that of all rectangles inscribed in a fixed circle, the square has the maximum area.

Let the circle have radius $r$. A rectangle inscribed in it has its diagonal equal to the circle's diameter $2r$. Let the rectangle have length $l$ and breadth $b$, so $l^2+b^2=4r^2$ (diagonal).

Write $l=2r\cos\theta,\ b=2r\sin\theta$ for some $\theta\in\left(0,\dfrac{\pi}{2}\right)$, so the constraint is automatically satisfied.

Area: $A(\theta)=l\cdot b=4r^2\sin\theta\cos\theta=2r^2\sin2\theta$.

$A$ is maximum when $\sin2\theta=1$, i.e. $2\theta=\dfrac{\pi}{2}\Rightarrow\theta=\dfrac{\pi}{4}$.

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