If y = x^x, prove that d^2y/dx^2 - 1/y (dy/dx)^2 - y/x = 0.

Question

If y = x^x, prove that =\(\cfrac{d^2y}{d\text x^2}-\cfrac1y\left(\cfrac{dy}{d\text x}\right)^2-\cfrac{y}{\text x}\) = 0.

d²y/dx² - 1/y (dy/dx)² - y/x

Answer 1

y = x^x

Taking log on both sides

log y = x log x

Differentiating with respect to x

Differentiating with respect to x

Hence Proved