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Let a function \( f: R \rightarrow R \) be given by \( f(x+y)=f(x) f(y) \) for all \( x, y \in R \) and \( f(x) \neq 0 \) for any \( x \in R \). If the function \( f(x) \) is differentiable at \( x=0 \), show that \( f^{\prime}(x)=f^{\prime}(0) f(x) \) for all \( x \in R \). Also, determine \( f(x) \).
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