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

Question

If $f(x+y)=f(x)\cdot f(y)\ \forall x,y$ and $f(5)=2,\ f'(0)=3$, then $f'(5)$ is

Accepted Answer

$$f\left( {x + y} \right) = f\left( x \right) \times f\left( y \right)$$

Differeniate with respect to $x,$ treating $y$ as constant

$$f'\left( {x + y} \right) = f'\left( x \right)f\left( y \right)$$

Putting $x=0$ and $y=x$, we get

$$f'\left( x \right) = f'\left( 0 \right)f\left( x \right);$$

$$ \Rightarrow f'\left( 5 \right) = 3f\left( 5 \right) = 3 \times 2 = 6.$$

Solutions · 1