**Introduction:**
In this problem, we are given a function f(x, y) = f(x) * f(y) and we are also given that f(1) = 2 for all x, y in R. We need to find the value of f'(1), i.e., the derivative of f(x) with respect to x at x = 1.

**Approach:**
To find the derivative f'(1), we need to differentiate the given function f(x, y) = f(x) * f(y) with respect to x and then evaluate it at x = 1.

**Differentiating f(x, y) with respect to x:**
To differentiate f(x, y) = f(x) * f(y) with respect to x, we can use the product rule of differentiation. According to the product rule, if we have a function g(x) = u(x) * v(x), then its derivative g'(x) is given by:

g'(x) = u'(x) * v(x) + u(x) * v'(x)

Applying the product rule to f(x, y) = f(x) * f(y), we get:

∂f/∂x = f'(x) * f(y) + f(x) * 0 (since f(y) is a constant with respect to x)

Simplifying, we get:

∂f/∂x = f'(x) * f(y)

**Evaluating the derivative at x = 1:**
Now, we need to evaluate the derivative ∂f/∂x at x = 1. Substituting x = 1 in the above equation, we get:

∂f/∂x |x=1 = f'(1) * f(y)

**Using f(1) = 2:**
We are also given that f(1) = 2 for all x, y in R. Substituting x = 1 in the original function, we get:

f(1, y) = f(1) * f(y)
2 = 2 * f(y)
f(y) = 1

**Substituting f(y) = 1:**
Now, substituting f(y) = 1 in the equation ∂f/∂x |x=1 = f'(1) * f(y), we get:

∂f/∂x |x=1 = f'(1) * 1
∂f/∂x |x=1 = f'(1)

Therefore, f'(1) = ∂f/∂x |x=1 = f'(1)

**Conclusion:**
After evaluating the derivative of f(x) with respect to x at x = 1, we find that f'(1) = f'(1). This implies that the value of f'(1) is equal to itself.