Question

Consider the non-constant differentiable function $f$ of one variable which obeys the relation $\frac{f\left(x\right)}{f\left(y\right)}=f(x-y).$ If $f^{\prime }\left(0\right)=p$ and $f^{\prime }\left(5\right)=q,$ then $f^{\prime }(-5)$ is

• A. $\frac{p^2}{q}$
• B. $\frac{q}{p}$
• C. $\frac{p}{q}$
• D. $q$

Answer 1

The correct option is A $\frac{p^2}{q}$

$f\left(x\right)=e^{kx},k\in R$ satisfies the condition.
$f^{\prime }\left(x\right)=k{e}^{kx}$
$f^{\prime }\left(0\right)=k=p$ and $f^{\prime }\left(5\right)=k{e}^{5k}=q$
$\therefore f^{\prime }(-5)=k{e}^{-5k}=\frac{p^2}{q}$