# Second order differential for change of variable?

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 $$x$$.

$$\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial x}\right)=\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial r}A\right)+\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial s}C\right)$$
$$=\frac{\partial^2 u}{\partial x\partial r}A+\underbrace{\frac{\partial u}{\partial r}\frac{\partial A}{\partial x}}_{=0 }+\frac{\partial^2 u}{\partial x\partial s}C+\underbrace{\frac{\partial u}{\partial s}\frac{\partial C}{\partial x}}_{=0 }$$
$$=\frac{\partial^2 u}{\partial x\partial r}A+\frac{\partial^2 u}{\partial x\partial s}C$$
$$\dfrac{\partial A}{\partial x}=\dfrac{\partial C}{\partial x}=0$$ since $$A$$ and $$C$$ are constants so when you take the second partial derivative of $$u$$ with respect to $$x$$ you will get
$$\frac{\partial^2u}{\partial x^2}=\frac{\partial^2u}{\partial x\partial r}A+\frac{\partial^2u}{\partial x\partial s}C$$