A function `f : R rarr R` satisfies the equation `f(x + y) = f(x) . f(y)` for all, `f(x) ne 0`. Suppose that the function is differentiable at x = 0 and f'(0) = 2. Then,

Options

A. f'(x) = 2f(x)
B. f'(x) = f(x)
C. f'(x) = f(x) + 2
D. f'(x) = 2f(x) + x

Answer

To solve the problem step by step, we will analyze the given functional equation and the conditions provided.

### Step 1: Analyze the Functional Equation
We are given that:
\[ f(x + y) = f(x) \cdot f(y) \]
for all \( x, y \in \mathbb{R} \) where \( f(x) \neq 0 \).