Can anyone explain this? Why the derivative with $x$ don't have lambda?

Question

I have difficulty in understanding the chain rule given in the picture. The derivative with $x$ do not have lambda, while $y$ term have. Why is it so?

Answer 1

You can write $\frac{\partial}{\partial x} = \frac{\partial \zeta}{\partial x}\frac{\partial}{\partial \zeta} + \frac{\partial \eta}{\partial x}\frac{\partial}{\partial \eta}$ and you can clearly see from starting expressions that $\frac{\partial \zeta}{\partial x} =\frac{\partial \eta}{\partial x} = 1 $. Similarly for y- derivative, which leads to $\lambda$ prefactors.