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).
Observe \begin{align} \frac{\partial}{\partial x} u = \left(\frac{\partial}{\partial \zeta}+\frac{\partial}{\partial \varepsilon} \right) u \end{align} then \begin{align} \frac{\partial^2}{\partial x^2} u =&\ \left(\frac{\partial}{\partial \zeta}+\frac{\partial}{\partial \varepsilon} \right) \left(\frac{\partial}{\partial \zeta}+\frac{\partial}{\partial \varepsilon} \right) u = \left(\frac{\partial}{\partial \zeta}+\frac{\partial}{\partial \varepsilon} \right)\left(\frac{\partial u}{\partial \zeta}+\frac{\partial u}{\partial \varepsilon}\right) \\ =&\ \frac{\partial}{\partial \zeta}\left(\frac{\partial u}{\partial \zeta}+\frac{\partial u}{\partial \varepsilon}\right)+\frac{\partial}{\partial \varepsilon}\left(\frac{\partial u}{\partial \zeta}+\frac{\partial u}{\partial \varepsilon}\right) \\ =&\ \frac{\partial^2u}{\partial\zeta^2}+2\frac{\partial^2 u}{\partial \zeta\partial \varepsilon} + \frac{\partial^2 u}{\partial \varepsilon^2} \end{align}