Here is some math copied from LaTeX:
$$\left(\frac{\partial P}{\partial \vec{V}}\right)_{P C, T C}=\left(\frac{\partial^{2} P}{\partial \vec{V}^{2}}\right)_{P C T C}=0$$
$$\ln(\frac{w^2\sqrt{[}4]t^3}{\sqrt{t+w}})$$
$$\ln(\frac{w^2\sqrt{[}4]t^3}{\sqrt{t+w}})$$