How do I expand the chain rule to second partial derivatives?

[rabbleguy]

We have f(x(y,z),t(y,z)).

This is more of a study question. I don't know how to expand out $$d^{2}f/dz^2$$


I know df/dz = df/dx*dx/dz + df/dt*dt/dz, but I don't know how to expand this to the 2nd derivative. I think the product rule comes into play? Not really sure.

Thanks for your help.

 

[LCKurtz]

Using the more convenient subscript notation, you have used:

f_z = f_xx_z+ f_tt_z

Now you have to do the same thing to this expression, which can be done termwise:

f_zz =(f_xx_z)_x$\cdot\,$x_z + (f_xx_z)_t$\cdot\,$t_z

and, yes, you will need the product rule inside those parentheses.