\(y = (x - 1) (5x^3 + \sqrt x)^\frac{11}2 \left(\frac{tanx - 1}{sin x}\right) e^{{5(x)}^\frac18} .\, x^{x^{x....}}\)
\(log \,y = log (x -1) + \frac{11}2 log (5x^3 + \sqrt x) + log(tan x - 1) - log \, sin x + 5(x) \frac 18 + x^{x^x} \)
On Differentiating both sides logx w.r.t x, we get
\(\frac1y \frac {dy}{dx} = \frac 1 {x- 1} +\frac{11}2\frac{15x^2 + \frac 1{2\sqrt x}}{5x^3+ \sqrt x} + \frac{sec^2 x}{tan x - 1} - \frac{cos x}{sin x} + \frac 58 x^{-\frac78} + \frac{x^{x^{x...}}}{x} + log x. \frac{x^{x^{x...}}}{x}\left(\frac 1{x^{x^{x...} }} - log x\right)\)
\(\therefore \frac{dy}{dx} = y\left(\frac 1{x - 1} + \frac {11}2 \frac{15x^2 + \frac1{2\sqrt x}}{5x^3 + \sqrt x} + \frac{sec^2x}{tan x -1} - cot x + \frac 58 \, \frac1{x^\frac 78} + \frac{x^{x^{x...}}}{x}\left(\frac {log x}{x^{x^{x..}}} - (log x)^2 + 1\right) \right)\)
y at x = 1 = 0
\(\therefore\frac{dy}{dx}|_{x=1} = 0\)