The problem shown in the image appears to be about proving a formula involving partial derivatives. The goal is to show that for a function $z$ which is a function of two variables $x$ and $y$, where $u$ and $v$ are new variables expressed as:

$u = ax + by \quad \text{and} \quad v = cx + dy,$

the relationship between the second partial derivatives of $z$ with respect to $x$ and $y$ can be written in a specific form. The objective is to prove:

$\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = \left( \frac{\partial u}{\partial x} \right)^2 \frac{\partial^2 z}{\partial u^2} + 2 \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} \frac{\partial^2 z}{\partial u \partial v} + \left( \frac{\partial v}{\partial x} \right)^2 \frac{\partial^2 z}{\partial v^2}$

To proceed with the proof, I would apply the chain rule to the transformation from variables $x$ and $y$ to $u$ and $v$, and compute the partial derivatives accordingly. Let's begin by first calculating the first-order derivatives and then proceed to the second-order derivatives.

Would you like a step-by-step explanation of how to perform these calculations, or do you want to explore a specific part of the solution?