Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying
\[\forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \quad \text{and} \quad \lim_{x \to 0} g(x)=1.\]
Find the derivative of $f$ in an arbitrary point $x.$
1. **Assume \( f(x) \neq 0 \) for all \( x \in \mathbb{R} \):**
   - If \( f(u) = 0 \) for some \( u \), then for any \( x \in \mathbb{R} \), we have:
     \[
     f(x) = f(x-u)f(u) = 0
     \]
     - This implies \( f(x) = 0 \) for all \( x \in \mathbb{R} \).
     - However, if \( f(x) = 0 \) for all \( x \), then \( f(x) = x g(x) + 1 \) would imply \( 0 = x g(x) + 1 \), leading to \( g(x) = -\frac{1}{x} \) for \( x \neq 0 \).
     - This contradicts the given condition \( \lim_{x \to 0} g(x) = 1 \).
   - Therefore, \( f(x) \neq 0 \) for all \( x \in \mathbb{R} \).

2. **Show \( f(x) > 0 \) for all \( x \in \mathbb{R} \):**
   - Since \( f(x) \neq 0 \) for all \( x \), and \( f(x) = f\left(\frac{x}{2}\right)^2 \), it follows that \( f(x) > 0 \) for all \( x \in \mathbb{R} \).

3. **Express \( f(x) \) in exponential form:**
   - Let \( f(x) = e^{h(x)} \). The functional equation \( f(x+y) = f(x)f(y) \) becomes:
     \[
     e^{h(x+y)} = e^{h(x)} e^{h(y)} \implies h(x+y) = h(x) + h(y)
     \]
   - This is the Cauchy functional equation, and its solutions are of the form \( h(x) = ax \) for some constant \( a \).

4. **Determine the constant \( a \):**
   - From the second given condition \( f(x) = x g(x) + 1 \), we have:
     \[
     e^{h(x)} = x g(x) + 1
     \]
   - Taking the limit as \( x \to 0 \):
     \[
     \lim_{x \to 0} \frac{e^{h(x)} - 1}{x} = \lim_{x \to 0} g(x) = 1
     \]
   - Using \( h(x) = ax \), we get:
     \[
     \lim_{x \to 0} \frac{e^{ax} - 1}{x} = a \cdot \lim_{x \to 0} \frac{e^{ax} - 1}{ax} = a \cdot 1 = a
     \]
   - Since \( \lim_{x \to 0} g(x) = 1 \), it follows that \( a = 1 \).

5. **Conclude the form of \( f(x) \):**
   - Therefore, \( h(x) = x \) and \( f(x) = e^x \).

6. **Find the derivative of \( f(x) \):**
   - The derivative of \( f(x) = e^x \) is:
     \[
     f'(x) = e^x
     \]

The final answer is \( \boxed{e^x} \).