Derivative on the Head.

$y=vx$

$y^d=v+x v^d$

$v+x v^d=v+\dfrac{\phi(v)}{\phi^{d}(v)}$

$\dfrac{\phi^{d}(v) dv}{\phi(v)}=\dfrac{dx}{x}$

$ln(\phi(v))=ln x+ln k$

$\phi(v)=kx$

$\phi(\dfrac{y}{x})=kx$

$y^\prime - \dfrac{y}{x} = \dfrac{\phi\left(\dfrac{y}{x}\right)}{\phi^\prime\left(\dfrac{y}{x}\right)}$

$\dfrac{xy^\prime - y}{x} = \dfrac{\phi\left(\dfrac{y}{x}\right)}{\phi^\prime\left(\dfrac{y}{x}\right)}$

$x\dfrac{xy^\prime - y}{x^2} = \dfrac{\phi\left(\dfrac{y}{x}\right)}{\phi^\prime\left(\dfrac{y}{x}\right)}$

$x = \dfrac{\phi\left(\dfrac{y}{x}\right)}{\phi^\prime\left(\dfrac{y}{x}\right) \left( \dfrac{xy^\prime - y}{x^2} \right)}$

$\dfrac{1}{x} = \dfrac{\phi^\prime\left(\dfrac{y}{x}\right) \left( \dfrac{xy^\prime - y}{x^2} \right)}{\phi\left(\dfrac{y}{x}\right)}$

$\dfrac{1}{x} = \dfrac{\left[\phi\left(\dfrac{y}{x}\right)\right]^\prime}{\phi\left(\dfrac{y}{x}\right)}$

$\left[ \ln (x) \right]^\prime = \left[\ln\left(\phi\left(\dfrac{y}{x}\right)\right)\right]^\prime$

$\left[ \ln\left(\phi\left(\dfrac{y}{x}\right)\right) - \ln (x) \right]^\prime = 0$

$\phi\left(\dfrac{y}{x}\right) \dfrac{1}{x} = k$

$\phi\left(\dfrac{y}{x}\right) = kx$