Function fusion #1

Calculus Level 3

Let \(f(x)=\int_{-1}^{x} e^{t^{2}} dt\) and \(h(x) = f(1+g(x))\) where \(g(x)\) is defined for all Real \(x\) and \(g'(x)\) exists for all Real \(x\). It is given that \(g(x)<0\) for all \(x>0\). If \(h'(1)=e\) and \(g'(1)=1\), then find \(g(1)\).

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