Step 1: Understanding the Concept:
The given functional equation \(f(x+y)=f(x)f(y)\) is the property of exponential functions. Functions that satisfy this are of the form \(f(x)=a^x\). We are asked to find the value of the derivative at a point, given the value of the function at that point and the value of the derivative at the origin.
Step 2: Key Formula or Approach:
1. Use the definition of the derivative for \(f'(x)\):
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
2. Apply the given functional equation \(f(x+h) = f(x)f(h)\) to simplify the expression for \(f'(x)\).
3. Use the given value of \(f'(0)\) to relate \(f'(x)\) to \(f(x)\).
4. Substitute \(x=3\) to find the final answer.
Step 3: Detailed Explanation:
Let's find the general expression for \(f'(x)\).
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
Using the functional equation, \(f(x+h) = f(x)f(h)\):
\[ f'(x) = \lim_{h \to 0} \frac{f(x)f(h) - f(x)}{h} = \lim_{h \to 0} \frac{f(x)(f(h) - 1)}{h} \]
Since \(f(x)\) does not depend on \(h\), we can take it out of the limit:
\[ f'(x) = f(x) \lim_{h \to 0} \frac{f(h) - 1}{h} \]
Now, let's look at the limit term. It looks like the definition of a derivative. Let's see what \(f'(0)\) is:
\[ f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0} \frac{f(h) - f(0)}{h} \]
From the original equation, \(f(x+y)=f(x)f(y)\), let \(x=x\) and \(y=0\). Then \(f(x) = f(x)f(0)\). Since this must hold for a non-trivial function \(f(x)\), we must have \(f(0)=1\).
(The question states \(f'(0)=11\), implying differentiability, which implies continuity. If \(f\) is continuous and not identically zero, then \(f(0)=1\).)
So, the derivative at 0 is:
\[ f'(0) = \lim_{h \to 0} \frac{f(h) - 1}{h} \]
This is exactly the limit term in our expression for \(f'(x)\). Therefore, we have the relationship:
\[ f'(x) = f(x) \cdot f'(0) \]
We are given \(f(3)=3\) and \(f'(0)=11\). We want to find \(f'(3)\).
Using the relationship we just derived:
\[ f'(3) = f(3) \cdot f'(0) \]
\[ f'(3) = 3 \cdot 11 = 33 \]
Step 4: Final Answer:
The value of \(f'(3)\) is 33, which corresponds to option (B).