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Find the derivative of f - [Iran Second Round 1985]

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December 28, 2010
calculusderivativefunctionlimitalgebra unsolvedalgebra

Problem Statement

Let f:R→Rf: \mathbb R \to \mathbb R and g:R→Rg: \mathbb R \to \mathbb R be two functions satisfying \forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases}   \text{and}   \lim_{x \to 0} g(x)=1. Find the derivative of ff in an arbitrary point x.x.
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