# Second Order Partial Derivative

I have a partial derivative: $$\frac{\partial u}{\partial x}= \frac{\partial u}{\partial \zeta}+\frac{\partial u}{\partial \varepsilon}$$ I want to find the second order partial derivative $\frac{\partial ^2u}{\partial x^2}$, and online I found the answer as: $$\frac{\partial ^2u}{\partial x^2}=\frac{\partial ^2u}{\partial \varepsilon^2}+2\frac{\partial ^2u}{\partial \varepsilon\zeta}+\frac{\partial ^2u}{\partial \zeta^2}$$ Can someone explain how they arrived at this?

If it's of any relevance, this is dealing with the 1-D wave equation and it's coming from here (equations 6 and 7).

• Thanks for the help! Just a question, what exactly does it mean to write $\frac {\partial} {\partial \zeta}$ on its own (i.e. $\frac {\partial} {\partial \zeta}$ + $\frac {\partial} {\partial \varepsilon}$)? I understand $\frac {\partial u} {\partial \zeta}$. – Andi Gu Apr 1 '17 at 4:37
• When I rewrite $\partial/\partial \zeta$ I am viewing the object as an operator which acts on functions $u$ (like a matrix acting on a vector, i.e. $A(x) = Ax$). – Jacky Chong Apr 1 '17 at 4:56