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  4. If f(x + y) = f(x) f(y) for all x and y and f(5) = 2, f'(0) = 3, then f'(5) is

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Q. If f(x + y) = f(x) f(y) for all x and y and f(5) = 2, f'(0) = 3, then f'(5) is

Limits and Derivatives

Solution:

$f\left(x+y\right) =f\left(x\right)f\left(y\right) $
$\Rightarrow f\left(x+5\right) =f\left(x\right)f\left(5\right)=2f\left(x\right) $
$\Rightarrow f'\left(x+5\right) =2f'\left(x\right)$
$ \Rightarrow f'\left(0+5\right) =2f'\left(0\right) =2\left(3\right) =6\Rightarrow f'\left(5\right) =6$