This question only regards to elementary manipulations with variables. I don't look into any technical details.
Consider two functions $F,f$ and two variables $s,x$. And a real number $E$. We have $$F(s) = Ef(x)$$ and $$s = Ee^x$$
I would like to calculate the $$\frac{d^2 F}{d s^2}$$ in terms of $f,x$ I know this may be simple, but only what i have got is the following ridiculous result: $$s^2 = E^2e^{2x}$$ $\frac{ds^2}{dx} = 2 E e^{2x}$, $\frac{dx}{dx^2}=\frac{1}{2 x}$. So we may multiply both sides to get: $$\frac{ds^2}{dx}\cdot \frac{dx}{dx^2} = \frac{ds^2}{dx^2} = \frac{E^2e^{2x}}{2 x}$$.
So we may put that altogether: $$\frac{d^F}{ds^2}=\frac{E d^2 f}{\frac{E^2 e^{2x}dx^2}{2x}}=\frac{2x d^2 f}{E e^{2x}dx^2}$$ Which it seems strange to me.
I hope you will help me.
Thank you in advance.
I would simply use the chain rule as $$\frac{dF}{ds}=Ef’(x)x_s$$ $$\frac{d^{2}F}{ds^2}=Ef’’(x){x_s}^2+Ef’(x)x_{ss}$$ and then use the fact $x_s s_x = 1$
For notation I am writing $x_s$ as the derivative of $x$ with respect to $s$.