You have a function of the form ##f(a,b,c)## where you are going to plug in ##a = y(x)##, ##b = y'(x)##, and ##c = x##. If you want to know how this changes when you let ##y(x) = y_0(x) + \epsilon y_1(x)##, you do the following:
$$
\frac{df}{d\epsilon} = \frac{\partial f}{\partial a} \frac{da}{d\epsilon} + \frac{\partial f}{\partial b} \frac{db}{d\epsilon} + \frac{\partial f}{\partial c} \frac{dc}{d\epsilon}.
$$
Here, ##a##, ##b##, and ##c## are just placeholders for the functions that are going to go into this. What ##\partial f/\partial a## really means is "the derivative of ##f## with respect to the first argument. In order not to have to write down dummy variables like this, we usually just write ##f(y,y',x)## and by ##\partial f/\partial y## we mean the derivative of ##f## with respect to its first argument and so on. Inserting ##a = y(x)##, ##b = y'(x)##, and ##c = x## gives you the correct result and ##da/d\epsilon = y_1##, ##db/d\epsilon = y'_1##, and ##dc/d\epsilon = 0##. The fact that ##y'_1## is not independent of ##y_1## encodes the fact that ##y'## is not independent of ##y##.
To take an analogy to usual multivariable calculus, consider the function ##f(x,y) = xy##. If we let ##x## and ##y## depend on a parameter ##t##, we find that
$$
\frac{df}{dt} = x' \partial_x f + y' \partial_y f = x' y + y' x.
$$
If we now assume that ##y = x^2##, we would find ##y' = 2xx'## and inserting this into the above would give
$$
\frac{df}{dt} = x' x^2 + 2xx' x = 3 x^2 x',
$$
the very same result as we would get if we had inserted ##y = x^2## into ##f## before differentiating, so it is consistent with ##y## being dependent on ##x##.