Question

Differentiate $y=x^{x^{...\infty }}$

• A. $\frac{dy}{dx}=\frac{y^2}{x(1-y\log x)}$
• B. $\frac{dy}{dx}=\frac{y}{x(1-y\log x)}$
• C. $\frac{dy}{dx}=\frac{y^3}{x(1-y\log x)}$
• D. None of these

Answer 1

Answer: (A)

$\frac{dy}{dx}=\frac{y^2}{x(1-y\log x)}$

$y=x^{x^{..\infty }}$

$y=x^y$

$\ln y=y\ln x$

Differentiating wrt $x$

$\Rightarrow \frac{y^{\prime }}{y}=\frac{y}{x}+y^{\prime }\ln x$

$y^{\prime }[1-y\ln x]=\frac{y^2}{x}$

$y^{\prime }=\frac{y^2}{x(1-y\ln x)}$