Differentiation and Invers become one

Calculus Level 4

F(x)=f(eg(x))\large F(x)=f\left( e^{g(x)} \right)

Let F(x)F(x) be defined as above, where f(x)=x3+5x+3 f(x)=x^3+5x+3 for 0x1 0 \leq x \leq 1 and g(x)=f1(x) g(x)=f^{-1}(x) . If the value of F(3)=ab {F}'(3)=\dfrac{a}{b} , where aa and bb are coprime positive integers, find a+b a+b .

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