Mathematics

Problem

Let

y=(\sin x)^{\cos x}

for

x\in\left(0,\dfrac{\pi}{2}\right)

. The value of

\dfrac{dy}{dx}

x=\dfrac{\pi}{4}

is a)

(\sin\dfrac{\pi}{4})^{\cos\dfrac{\pi}{4}}\left[\dfrac{\sqrt{2}}{4}\,\ln 2+\dfrac{\sqrt{2}}{4}\right]

(\sin\dfrac{\pi}{4})^{\cos\dfrac{\pi}{4}}\left[\dfrac{\sqrt{2}}{4}\,\ln 2+\dfrac{1}{\sqrt{2}}\right]

(\sin\dfrac{\pi}{4})^{\cos\dfrac{\pi}{4}}\left[-\dfrac{\sqrt{2}}{4}\,\ln 2+\dfrac{1}{\sqrt{2}}\right]

(\sin\dfrac{\pi}{4})^{\cos\dfrac{\pi}{4}}\left[\dfrac{\sqrt{2}}{4}\,\ln 2+1\right]