We know that f'(x) and f"(x) respectively denote the first derivative and second derivative of a given function f(x) with respect to x.

If P is the inverse function of S, and P'(x)=e

^{x2}; then:

2*S(x)*S'(x) + e^{S(x)2}*S"(x) = ?

I wish to acknowledge the superb analytical prowess of JayDeeKay whose short yet comprehensive methodology has forced me to replace my solution with his resolution of the problem under reference.