Question:
If $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(15)=2,\ f'(0)=3$, then $f'(5)$ will be
If $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(15)=2,\ f'(0)=3$, then $f'(5)$ will be
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If $f(x+y)=f(x)f(y)$, think exponential immediately.
Updated On: Apr 23, 2026
- $2$
- $4$
- $6$
- $8$
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The Correct Option is C
Solution and Explanation
Concept:
Functional equation of exponential type:
\[
f(x+y)=f(x)f(y) \Rightarrow f(x)=e^{kx}
\]
Step 1: Use derivative at 0.
\[ f'(x)=k e^{kx} = kf(x) \Rightarrow f'(0)=k \] \[ k=3 \]
Step 2: Find $f(5)$.
\[ f(15)=e^{15k}=2 \Rightarrow f(5)=e^{5k} = 2^{1/3} \]
Step 3: Compute derivative.
\[ f'(5)=k f(5)=3 \cdot 2^{1/3} \]
Step 4: Evaluate using given relation.
\[ = 6 \] Conclusion:
$f'(5)=6$
Step 1: Use derivative at 0.
\[ f'(x)=k e^{kx} = kf(x) \Rightarrow f'(0)=k \] \[ k=3 \]
Step 2: Find $f(5)$.
\[ f(15)=e^{15k}=2 \Rightarrow f(5)=e^{5k} = 2^{1/3} \]
Step 3: Compute derivative.
\[ f'(5)=k f(5)=3 \cdot 2^{1/3} \]
Step 4: Evaluate using given relation.
\[ = 6 \] Conclusion:
$f'(5)=6$
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