Let `f` be a function such that `f(x+y)=f(x)+f(y)` for all `xa n dya n df(x)=(2x^2+3x)g(x)` for all`x ,` where `g(x)` is continuous and `g(0)=3.` Then find `f^(prime)(x)dot`

Explanation

To solve the problem step by step, we will analyze the given conditions and derive the required derivative \( f'(x) \).

### Step 1: Use the given functional equation
We know that:
\[ f(x+y) = f(x) + f(y) \]
This is a Cauchy functional equation, which implies that \( f(x) \) can be expressed in the form \( f(x) = kx \) for some constant \( k \) if we assume \( f \) is continuous.
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