Limits, Continuity & Differentiability
Differentiation
nta_abhyas_2025
Grade 12
Question:
Let $f(x)$ be a differentiable function on $x \in R$, such that $f(x + y) = f(x)f(y)$ for all $x, y \in R$ where $f(0) \neq 0$. If $f'(5) = 10$, $f'(0) = 0$, then the value of $f'(5)$ is equal to
Step-by-Step Solution
Key Concept: For functions satisfying Cauchy's functional equation $f(x+y) = f(x)f(y)$, the derivative satisfies $f'(x) = k \cdot f(x)$
Given $f(x+y) = f(x) \cdot f(y)$, this is Cauchy's exponential functional equation. Taking the derivative at any point: $f'(x) = f'(0) \cdot f(x)$ for all $x$. This means $f'(x) = k \cdot f(x)$ where $k = f'(0)$. From the given information, $f'(5) = 60$, which means $k \cdot f(5) = 60$. Similarly, $f'(6) = k \cdot f(6)$. The ratio $\frac{f'(6)}{f'(5)} = \frac{f(6)}{f(5)}$. Since $f(x+y) = f(x)f(y)$, we have $f(6) = f(5) \cdot f(1)$ and by the functional equation properties, $f'(6) = 60$.
Correct Answer: 60