If y = x^x , then value of displaystyledy/dx at x = 2 is:

The correct answer is Option (2) → \(4(1+\log 2)\)

Both the base and the power contain x, so we take logarithms (logarithmic differentiation):

    \(\log y = x\log x\)

Differentiating with respect to x:

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

    \( \displaystyle \frac{dy}{dx} = x^x(1 + \log x) \)

At x = 2:

    \( \displaystyle \frac{dy}{dx} = 2^2(1 + \log 2) \)

    \(= 4(1 + \log 2)\)