Question

Let $f\left(x\right)$ be a differentiable function which satisfies the equation $f\left(xy\right)=f\left(x\right)+f\left(y\right)$ for all $x>0$,$y>0$

then $f^{\prime }\left(x\right)$ is equal to

• A. $\frac{f^{\prime }\left(x\right)}{x}$
• B. $\frac{1}{x}$
• C. $f^{\prime }\left(x\right)$
• D. $f^{\prime }\left(x\right).\left(\ln x\right)$

Answer 1

The correct option is A $\frac{f^{\prime }\left(x\right)}{x}$

$f\left(xy\right)=f\left(x\right)\ne f\left(y\right)$

Partial differentiation w.r.t. x

$\partial \frac{f\left(xy\right)}{\partial x}=\frac{\partial f\left(x\right)}{\partial x}+\frac{\partial f\left(y\right)}{\partial x}$

$=\left(f^{{}^{\prime }}\left(xy\right)\right)\times (y\cdot 1)=f^{{}^{\prime }}\left(x\right)+0$

Put $y=\frac{1}{x}$

$\Rightarrow f^{{}^{\prime }}\left(x\right)=f^{{}^{\prime }}(x\times \frac{1}{x})\times \left(\frac{1}{x}\right)$

$=\frac{f^{{}^{\prime }}\left(x\right)}{x}$