I need help spotting a corresponding transformation
Let $x,y$ be some variables and $$z=z(x,y)$$. We have a transformation $X(\lambda):(x,y,z)\to (x',y',z')$, such that $$x'= x\exp(a\lambda)\\ y'= y\exp(-a\lambda)\\ z'= z\exp(-3a\lambda)$$ where $a$ is a constant. Let $$w\equiv z^2-\frac{\partial z}{\partial x}$$ What transformation $X'(\lambda)$ such that $X'$ transforms $(x,y,w)\to (F_x(x,y,w),F_y(x,y,w), F_w(x,y,w))$ to the corresponding quantities given by $X(\lambda)$ where $F_i$ are functions of $x,y,w$?
By corresponding I just mean that the "values/expressions" of $$X'(x),X'(y),X'(w)$$ are equal to $$X(x),X(y),X\left(z^2-\frac{\partial z}{\partial x}\right)$$ respectively. Note though that $X'(x),X'(y),X'(w)$ are functions of $x,y,w$
I presume the action of $X'(\lambda)$ on $x,y$ remains the same. But what about that on $w$?
We have $$w'= z'^2-\frac{\partial z'}{\partial x'}\\ = z^2\exp(-6a\lambda)-\exp(-4a\lambda)\frac{\partial z}{\partial x}$$ But how can I express this in terms of $x,y,w$?
Advanced thanks for any help!