Formula used : log mn = n log mDifferentiation by part: \(\frac{d}{dx}(uv) = v\frac{du}{dx}+u\frac{dv}{dx}\)\(\frac{d}{dx}x^n=nx^{n-1}\)\(\frac{d}{dx}log\ x= \frac{1}{x}\) Calculation :Given that y = (xx)x,Taking log on both sideslog y = log (xx)xUsing the formula (1)? log y = x log xx? log y = x2 log xDiff. with respect to x \(\frac{d}{dx}\)(log y) = \(\frac{d}{dx}\)(x2log x)Doing diff. by part as discussed in formula (2)? \(\frac{1}{y}\frac{dy}{dx} = x^2\frac{d}{dx}log\ x+ log \ x\frac{d}{dx}x^2\)Since, \(\frac{d}{dx}x^n=nx^{n-1}\) & \(\frac{d}{dx}log\ x= \frac{1}{x}\)? \(\frac{1}{y}\frac{dy}{dx} = \frac{x^2}{x}+2x\ log \ x\)? \(\frac{1}{(x^x)^x}\frac{dy}{dx} = \frac{x^2}{x}+2x\ log \ x\) (? y = (xx)x)Put x = 1 in above equation? \(\frac{dy}{dx} \) = 1 + 2 × 1 log 1? \(\frac{dy}{dx} \) = 1 (? log 1 = 0)? \(\frac{dy}{dx}\) at x = 1 is 1.