Let $f(x y)=x f(y)+y f(x)$ for all $x, y \in R_{+}$and $f(x)$ be differentiable in $(0, \infty)$, th

Question

Let $f(x y)=x f(y)+y f(x)$ for all $x, y \in R_{+}$and $f(x)$ be differentiable in $(0, \infty)$, then determine $f(x)$.

Professor PC Thomas Classes · Answer

Given $f(xy)=xf(y)+yf(x)$
Differentiating both sides w.r.t. x treating y as constant, $f^{\prime}(xy).y=f(y)+yf^{\prime}(x)$
Putting $y=x$ and $x=1$, then
$f^{\prime}(x)x=f(x)+xf^{\prime}(1)$
$\Rightarrow\frac{xf^{\prime}(x)-f(x)}{x^{2}}=\frac{f^{\prime}(1)}{x} \Rightarrow\frac{d}{dx}(\frac{f(x)}{x}=\frac{f^{\prime}(1)}{x})$
Integrating both sides w.r.t. x taking limits 1 to x,
$\frac{f(x)}{x}-\frac{f(1)}{1}=f^{\prime}(1)\{ln~x-ln~1\} \Rightarrow\frac{f(x)}{x}-0=f^{\prime}(1)ln~x(\cdot\cdot\cdot f(1)=0)$
Hence, $f=f^{\prime}(1)(x~lnx)$