I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow
To be more specific, it's the diferentiation of (6.9) right at the top of page 220
The way I see it, shouldn't it have been
$$\frac{\partial f}{\partial (y+ \eta \alpha)} \frac{\partial (y+ \eta \alpha)}{\partial \alpha} + ... = \frac{\partial f}{\partial (y+ \eta \alpha)} \eta + ...$$
Instead, he differentiated w.r.t. $y$ instead of $(y+ \eta \alpha)$, why is that?
If we have a 3-place function $f(y,y',x)$ it is common notation to use $\frac{\partial f}{\partial y}$, $\frac{\partial f}{\partial y'}$ and$\frac{\partial f}{\partial x}$ to mean respectively the first, second and third partial derivatives.
In the example you provide, it is taking the first and second derivatives wrt $\alpha$ (the third does not appear since the third argument of the function does not variate in function of $\alpha$).
Sorry for the typo in the comment, I meant $\partial$ and not $\delta$.