Notation on partial deriviatives

Question

If I need to find $$\frac{\partial ^{2}g}{\partial u \partial v}$$

Then do I want to perform $$ \frac{\partial} { \partial v}\ \big( \frac{\partial g}{\partial u} \big) $$ or $$ \frac{\partial g} {\partial u} * \frac{\partial g}{\partial v} $$

I'm thinking the first one, but I just want to check my intuition.

edit: Notation Blunder *facepalm *

Answer 1

If you intended to write $\frac{\partial^2 g}{\partial u\partial v}$, then $$ \frac{\partial^2 g}{\partial u\partial v}= \frac{\partial}{\partial u}\left(\frac{\partial g}{\partial v} \right)= \frac{\partial}{\partial v}\left(\frac{\partial g}{\partial u} \right) $$

Answer 2

$$\frac{\partial g^{2}}{\partial u \partial v}$$ has practically no meaning, or it's a bad notation that implies $$\frac{\partial^2 g}{\partial u \partial v}$$ which means $$\frac{\partial} { \partial v}\ \left( \frac{\partial g}{\partial u} \right)$$