Troubleshooting Calculations for Equations y=vx and dy/dx=v+x*dv/dx

[asdf1]

for this equation, y=vx, it's the same as v=y/x
however, if i differenate the 1st equation, dy/dx=v+x*dv/dx
and if i differenate the 2nd equation, dv/dx=(1/x)*dy/dx + [-1/(x^2)]y
but both differenated equations don't seem the same...
does somebody know where my calculuations went wrong?

 

[Astronuc]

dv/dx=(1/x)*dy/dx + [-1/(x^2)]y

becomes x (dv/dx) = dy/dx - y/x, but v = y/x

so x (dv/dx) = dy/dx - v

or

x (dv/dx) + v = dy/dx, which is same as other equation.

 

[asdf1]

thank you very much! :)