# Second order partial derivatives

Suppose that $$z=g(x,y),\, x=s+t,$$ and $$y=st$$, where all first and second order partial derivatives of $$g$$ exist and are continuous.

Show that $$\frac{\partial ^2 z}{\partial s\partial t}=\frac{\partial ^2 g}{\partial x^2}+x\frac{\partial^2g}{\partial x\partial y}+y\frac{\partial^2 g}{\partial y^2}+\frac{\partial g}{\partial y}.$$

Someone told me I have to use the chain rule twice, but I still don't quite understand what I'm meant to do.

• Well, I expanded all the second order terms (e.g. the term on the LHS would be d/ds(dz/dt), and I assumed that dg/dy = dz/dy since z = g(x,y). And I found the partial derivatives dx/ds, dx/dt, dy/ds, dy/dt etc, but I'm having trouble understanding how to put it all together
– user190322
Nov 6, 2014 at 11:51
• Could you edit the question to show how you expanded $\frac{\partial^2g}{\partial x^2}$ for example? It will give us a better look on the troubles you're having and it will likely produce more helpfull answers. Nov 6, 2014 at 11:57
• If you are having trouble formatting the maths, there is a tutorial here on how to format on this site. Nov 6, 2014 at 12:00

EDIT: @Kevin is correct, I needed to replace full derivaties w.r.t $s, t$ with partials. The remaining parts of the answer is correct. I would reply to Kevin as a comment about this but the "add comment" function on my PC seems to be disabled.