Rearranging separable equations

[thedude36]

I'm having difficulty with the question in the pic provided. (http://i.imgur.com/Fg7CHoY.png). Basically the equation I am given is

dy/dx = (y-4x)/(x-y)​

and it walks through the steps needed to solve it, however, I am supposed to show how to arrive at each step given. I've gotten part a, where it asks to show that the above equation can be written as

dy/dx = (y/x - 4)/(1-y/x)​

and, introducing v=y/x, rewrite dy/dx in terms of v, x, and dv/dx. What I get for part b is

dy/dx = d(xv)/dx = x(dv/dx)​

however, on part c, it looks like I'm supposed to get

dy/dx = v+x(dv/dx)​

I'm not sure where the additional v term comes from. If anyone could help me out, i would really appreciate it!

 

[haruspex]

dy/dx = d(xv)/dx = x(dv/dx)
dy/dx = v+x(dv/dx)​

It's just the product rule. How do you differentiate f(x)g(x)?

 

[thedude36]

I can't believe it was that simple - I had been staring at it for hours. Thank you!