I have a question about partial differential equations How to find $\frac{\partial^2 e^u}{\partial y^2} =$? When $u = u(x,y)$

$$\frac{\partial}{\partial y}e^u=e^u\frac{\partial u}{\partial y},$$
$$\frac{\partial^2}{\partial y^2}e^u
=\frac{\partial}{\partial y}\left(e^u\frac{\partial u}{\partial y}\right)
=\left(\frac{\partial}{\partial y}e^u\right)\frac{\partial u}{\partial y}+e^u\frac{\partial}{\partial y}\frac{\partial u}{\partial y}
=e^u\left(\frac{\partial u}{\partial y}\right)^2+e^u\frac{\partial^2u}{\partial y^2}.$$

Or more concisely,
$$\frac{\partial}{\partial y}e^u=e^uu_y,$$
$$\frac{\partial^2}{\partial y^2}e^u=\frac{\partial}{\partial y}\left(e^uu_y\right)=\left(\frac{\partial}{\partial y}e^u\right)u_y+e^u\frac{\partial}{\partial y}u_y=e^uu_y^2+e^uu_{yy}.$$
