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A function f: RvecR satisfies the equati...

A function `f: RvecR` satisfies the equation `f(x+y)=f(x)f(y)` for all`x , y in Ra n df(x)!=0fora l lx in Rdot` If `f(x)` is differentiable at `x=0a n df^(prime)(0)=2,` then prove that `f^(prime)(x)=2f(x)dot`

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To solve the problem step by step, we will start with the given functional equation and use the properties of differentiability. ### Step 1: Analyze the functional equation We are given that: \[ f(x+y) = f(x)f(y) \] for all \( x, y \in \mathbb{R} \). ### Step 2: Find \( f(0) \) ...
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