Second order differential for change of variable?

Question

I have $r=Ax+Bt$ and $s=Cx+Dt$

I know $\frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}A+\frac{\partial u}{\partial s}C$

But I don't understand how to differentiate this again with respect to $x$.

Answer 1

$$\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial x}\right)=\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial r}A\right)+\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial s}C\right)$$

$$=\frac{\partial^2 u}{\partial x\partial r}A+\underbrace{\frac{\partial u}{\partial r}\frac{\partial A}{\partial x}}_{=0 }+\frac{\partial^2 u}{\partial x\partial s}C+\underbrace{\frac{\partial u}{\partial s}\frac{\partial C}{\partial x}}_{=0 }$$

$$=\frac{\partial^2 u}{\partial x\partial r}A+\frac{\partial^2 u}{\partial x\partial s}C$$

Answer 2

$\dfrac{\partial A}{\partial x}=\dfrac{\partial C}{\partial x}=0$ since $A$ and $C$ are constants so when you take the second partial derivative of $u$ with respect to $x$ you will get

$$\frac{\partial^2u}{\partial x^2}=\frac{\partial^2u}{\partial x\partial r}A+\frac{\partial^2u}{\partial x\partial s}C$$