If $y=x^{x^{x^{\dots }}}$, then $\frac{dy}{dx}$ is equal to

Solution

1. Given the infinite power tower $y=x^{x^{x^{\dots }}}$, we can rewrite the expression as $y=x^y$.
2. Taking the natural logarithm of both sides gives $\ln y=y\ln x$.
3. Differentiating both sides with respect to $x$ using the product rule yields $\frac{1}{y}\frac{dy}{dx}=\frac{dy}{dx}\ln x+\frac{y}{x}$.
4. Rearranging the terms to isolate $\frac{dy}{dx}$ gives $\frac{dy}{dx}(\frac{1}{y}-\ln x)=\frac{y}{x}$, which simplifies to $\frac{dy}{dx}=\frac{y^2}{x(1-y\ln x)}$.
5. Hence the answer is (A).