Q67 / 155 · WBJEE 2017 · Mathematics
Consider the non-constant differentiable function $f$ of one variable which obeys the relation $\frac{f(x)}{f(y)}=f(x-y) .$ If $f^{\prime}(0)=p$ and $f^{\prime}(5)=q,$ then $f^{\prime}(-5)$ is
- Correct answerA$\frac{p^{2}}{q}$
- B$\frac{q}{p}$
- C$\frac{p}{q}$
- D$q$
Solution
We have, $\frac{f(x)}{f(y)}=f(x-y)$
$\Rightarrow \quad f(x)=a^{k x}$
$\therefore \quad f^{\prime}(x)=k a^{b x} \log a$
$\begin{array}{lr}\text { Again, } & f^{\prime}(0)=P \\ \Rightarrow & k a^{\circ} \log a=P \\ \Rightarrow & k \log a=P\end{array}$
Also, $\quad f^{\prime}(5)=q$
$\Rightarrow \quad k a^{5 k} \log a=q$
$a^{5 k} p=q$
$a^{5 k}=\frac{q}{P}$
Now, $\begin{aligned} f^{\prime}(-5) &=k a^{-4} \log a \\ &=\frac{k \log a}{a^{\Delta}} \end{aligned}$ $=\frac{P}{\left(\frac{q}{p}\right)}=\frac{p^{2}}{q}$
Hence, the correct answer is option: 'A'.