Infinite derivative of nested radicals

Question

$$\cdots \frac{d}{dx}\frac{d}{dx}\frac{d}{dx} \sqrt{x+\sqrt[3]{x+\sqrt[4]{x\cdots}}}$$ Its not super hard to find a finite number of derivatives, but I can not understand how to pull off infinite here. Please help. Thank you.(Is it even logical?)

Answer 1

Consider the function \begin{align} S_{n}(x) = \sqrt[2]{x + \sqrt[3]{x + \sqrt[4]{x + \cdots + \sqrt[n+2]{x}}}} \end{align} For the case of $S_{2}(x)$ then \begin{align} D S_{2}(x) &= D\left[ \sqrt[2]{x + \sqrt[3]{x + \sqrt[4]{x}}} \, \right] \\ &= \frac{1}{2 \, S_{2}(x) } \, \left[ 1 + \frac{1}{3 \, \left( \sqrt[3]{x + \sqrt[4]{x}} \right)^{2}} \, \left( 1 + \frac{1}{4 \, \sqrt[4]{x^{3}}} \right) \right] \end{align} Further differentiation can be applied. If this problem was found in a book the line "...an exercise left for the reader." is applied.