Why Is u=y/x Treated as a Function of x Alone in ODE Differentiation?

[KT KIM]

I am studying ode now, and my text has that
If y'=f(y/x)
Then, setting y/x=u ; y=ux is a way to solve it.
I understand the idea, turn orignal form to separable form.

But I can't get the differentiation, Book says
y'=u'x+u by product rule which I already know.
Here my question is why u=y/x that obviously has two variables x & y, u(x,y) should be differentiated respect to x like it only has one variable x ( like u(x) )

 

[Mark44]

I am studying ode now, and my text has that
If y'=f(y/x)
Then, setting y/x=u ; y=ux is a way to solve it.
I understand the idea, turn orignal form to separable form.

But I can't get the differentiation, Book says
y'=u'x+u by product rule which I already know.
Here my question is why u=y/x that obviously has two variables x & y, u(x,y) should be differentiated respect to x like it only has one variable x ( like u(x) )

x is being differentiated.
Starting with y = ux, we differentiate everything with respect to x.
y' = ux' + u'x
Here, x' means $\frac{d}{dx}x$, which simplifies to 1, leaving us with $y' = u \cdot 1 + u'x = u + u'x$.