I want to differentiate $$y=\exp(x+\exp(x+\exp(x+\exp(x+\cdots)))).$$
Is the following substitution and differentiation appropriate?
\begin{align} y&=\exp(x+\exp(x+\exp(x+\exp(x+\cdots))))\\ y&=\exp(x+y)\\ \frac{dy}{dx}&=\exp(x+y)\left(1+\frac{dy}{dx}\right)\\ \frac{dy}{dx}&=y\left(1+\frac{dy}{dx}\right)\\ \frac{dy}{dx}&=\frac{y}{1-y}\\ \end{align}
As well, I notice that I can write the derivative (and therefore any higher order derivative) in terms of $y$ only. What is required of the $\exp$ function such that I will not be able to write the derivative in terms of $y$ only? It does not seem like the reason is that the $x$ in the argument is just $x$ and not some function of $x$: for example, \begin{align} y&=\exp x^2\qquad\quad\implies x=\log y-\log2\\ \frac{dy}{dx}&=2x\exp x^2\\ \frac{dy}{dx}&=2xy\\ \frac{dy}{dx}&=2y\,(\log y-\log2)\\ \end{align} and now the derivative is free of $x$. Knowing when this is the case would be useful in constructing problems where we can find the slope of the tangent line given only a $y$ value.