If y=x^(xsin x) then displaystyledy/dx=?

The correct answer is Option (3) → \(x^{(x\sin x)}[\sin x\,(1+\log x)+x\log x\cos x]\)

Both the base and the power contain x, so we use logarithmic differentiation (x > 0):

    \(\log y=x\sin x\,\log x\)

The right-hand side is a product of three functions x, \(\sin x\) and \(\log x\). Differentiating each in turn (product rule):

    \( \displaystyle \frac{1}{y}\frac{dy}{dx}=1\cdot\sin x\log x+x\cos x\log x+x\sin x\cdot\frac{1}{x} \)

    \( \displaystyle =\sin x\log x+x\log x\cos x+\sin x \)

    \( \displaystyle =\sin x\,(1+\log x)+x\log x\cos x \)

Multiplying by \(y=x^{(x\sin x)}\):

    \( \displaystyle \frac{dy}{dx}=x^{(x\sin x)}[\sin x\,(1+\log x)+x\log x\cos x] \)

Note: NTA dropped this question. For the benefit of students and to give them additional practice, the question has been suitably modified so that it has one clear answer.