For $ z=h(x,y) $, with $x=2s^2+3t$ and $y=2s-t^2$, I need to find $$ \frac{\partial^2z}{\partial s\partial t}$$
I know that $\frac{\partial x}{\partial s} = 4s$, $\frac{\partial x}{\partial t} = 3$ $\frac{\partial y}{\partial s} = 2$ $\frac{\partial y}{\partial t} = -2t$.
Now, $\frac{\partial^2z}{\partial s\partial t}=\frac{\partial}{\partial s}\frac{\partial z}{\partial t}$ and I know that $\frac{\partial z}{\partial t}=3h_1 (x,y)-2th_2 (x,y)$
But then, I get confused with the order of symbols for the following:
$$ \frac{\partial}{\partial s} (3h_1 (x,y)-2th_2 (x,y)) $$
Which order of subscripts should I take to differentiate with respect to s?
Is the following correct?
$$\frac{\partial^2z}{\partial s\partial t}= 3\left(h_{11}(x,y)\frac{\partial x}{\partial s}+h_{12}(x,y)\frac{\partial y}{\partial s}\right)-2t\left(h_{12}(x,y)\frac{\partial x}{\partial s}+h_{22}(x,y)\frac{\partial y}{\partial s}\right) $$
For the sake of consistency: $$\frac{\partial^2z}{\partial s\partial t}= 3\left(h_{11}(x,y)\frac{\partial x}{\partial s}+h_{12}(x,y)\frac{\partial y}{\partial s}\right)-2t\left(h_{\color{red}{21}}(x,y)\frac{\partial x}{\partial s}+h_{22}(x,y)\frac{\partial y}{\partial s}\right) $$
Otherwise it's correct, since $h_{12}=h_{21}$.