This is how I personally pronounce them:

  1. I pronounce it either "dee wai over dee eks" or simply "dee wai dee eks".
  2. I occasionally pronounce it as "dee squared wai over dee eks squared", but more often I just refer to it as "the second derivative of y with respect to x".
  3. "Partial of y with respect to x." Very occasionally as "del wai del eks".
  4. As with 2, I usually just call this "the second partial of y with respect to x".
  5. "del cubed y over del $x^2$ del z", but if we have $y=f(x,z)$, I'd much prefer referring to it as $f_{zxx}$.

3. $$ \frac{\partial y}{\partial x}\rightarrow\text{day wai, day eks}$$

5. $$ \frac{\partial^3 y}{\partial x^2\partial z}\rightarrow\text{day cubed wai, day eks squared, day zee}$$

6. $$\nabla F\rightarrow\text{del eff}$$