Suppose I have a function: $f(x(\eta),\eta)$ and I want to take the derivative with respect to $\eta$. Note that $f$ is a function of $x$ and $\eta$ and that $x$ itself is a function of $\eta$.
I am getting that the derivative of this function would look like: $\displaystyle \frac{\partial f}{\partial x}\frac{dx}{d\eta}+\frac{\partial f}{\partial \eta}$
Does this look correct or am I way off?
That's right, since $$\frac{\mathrm d\eta}{\mathrm d\eta}=1.$$