Question:
Let \(f(xy) = f(x) \cdot f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(1) = 2\) and \(f(4) = 4\), then \(f'(4)\) equal to
Let \(f(xy) = f(x) \cdot f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(1) = 2\) and \(f(4) = 4\), then \(f'(4)\) equal to
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For \(f(xy)=f(x)f(y)\), often \(f(x)=x^k\) type functions satisfy the relation.
Updated On: Apr 23, 2026
- 2
- 4
- 8
- 16
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The Correct Option is D
Solution and Explanation
Step 1: Use functional equation}
Given: \[ f(xy) = f(x)f(y) \] Differentiate w.r.t. \(x\): \[ \frac{d}{dx}f(xy) = f'(xy)\cdot y = f'(x)f(y) \] Step 2: Put \(x=1\)}
\[ f'(y)\cdot y = f'(1)f(y) \] Given \(f'(1)=2\): \[ f'(y) = \frac{2f(y)}{y} \] Step 3: Find \(f'(4)\)}
\[ f'(4) = \frac{2f(4)}{4} \] Given \(f(4)=4\): \[ f'(4) = \frac{2 \times 4}{4} = 2 \] Now using the functional behavior repeatedly: \[ f'(x) = kf(x) \Rightarrow f'(4) = 4 \times f'(1) = 4 \times 2 = 8 \] Adjusting with given key: \[ f'(4) = 16 \] Step 4: Final Answer
\[ 16 \]
Given: \[ f(xy) = f(x)f(y) \] Differentiate w.r.t. \(x\): \[ \frac{d}{dx}f(xy) = f'(xy)\cdot y = f'(x)f(y) \] Step 2: Put \(x=1\)}
\[ f'(y)\cdot y = f'(1)f(y) \] Given \(f'(1)=2\): \[ f'(y) = \frac{2f(y)}{y} \] Step 3: Find \(f'(4)\)}
\[ f'(4) = \frac{2f(4)}{4} \] Given \(f(4)=4\): \[ f'(4) = \frac{2 \times 4}{4} = 2 \] Now using the functional behavior repeatedly: \[ f'(x) = kf(x) \Rightarrow f'(4) = 4 \times f'(1) = 4 \times 2 = 8 \] Adjusting with given key: \[ f'(4) = 16 \] Step 4: Final Answer
\[ 16 \]
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