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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?

Show that for a function z of two variables x and y, with u and v defined as u = ax + by and v = cx + dy, the second-order partial derivatives of z satisfy the given equation.

Explore the step-by-step proof of transforming second-order partial derivatives of z with respect to x and y using the chain rule. This multivariable calculus problem involves defining new variables u and v and deriving their relationships.

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