Explanation

(c)
$\because y = x^{In\ x}$
On taking log on both sides, we get
In $y = (In\ x)^{2}$
On differentiating w.r.t. $x,$we get
$\frac{1}{y}\frac{dy}{dx} = \frac{2\ In\ x}{x}$
$\Rightarrow \frac{dy}{dx} = y\frac{2\ In\ x}{x} = \frac{2\left( x^{In\ x} \right)In\ x}{x}$
$\Rightarrow \frac{dy}{dx} = 2x^{In\ x - 1}In\ x$