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## Main Question or Discussion Point

I was wondering how to prove the multivariable chain rule

[tex]\frac{\mbox{d}z}{\mbox{d}t}=\frac{\partial z}{\partial y}\frac{\mbox{d}y}{\mbox{d}t}+\frac{\partial z}{\partial x}\frac{\mbox{d}x}{\mbox{d}t}[/tex]

where [tex]z=z(x(t),y(t))[/tex]

I don't really need an extremely rigorous proof, but a slightly intuitive proof would do.

Also how does one prove that if z is continuous, then

[tex]\frac{{\partial}^{2}z}{\partial x \partial y}=\frac{{\partial}^{2}z}{\partial y \partial x}[/tex]

Thanks in advance.

[tex]\frac{\mbox{d}z}{\mbox{d}t}=\frac{\partial z}{\partial y}\frac{\mbox{d}y}{\mbox{d}t}+\frac{\partial z}{\partial x}\frac{\mbox{d}x}{\mbox{d}t}[/tex]

where [tex]z=z(x(t),y(t))[/tex]

I don't really need an extremely rigorous proof, but a slightly intuitive proof would do.

Also how does one prove that if z is continuous, then

[tex]\frac{{\partial}^{2}z}{\partial x \partial y}=\frac{{\partial}^{2}z}{\partial y \partial x}[/tex]

Thanks in advance.