Partial derivatives chain rule clarification needed

Question

In this example here:

I am having trouble understanding the steps in applying the product rule, like what is set as u, du, v, du?

Please help clarify.

Answer 1

Hint:

For the first product: $$ \frac{\partial}{\partial x} x^3 z^2=z^2\frac{\partial}{\partial x}(x^3)+x^3\frac{\partial}{\partial x}(z^2)=z^2(3x^2)+x^3\left(2z\frac{\partial}{\partial x}z\right) $$ You can termite. And use the same rule for the other product.