Calculating derivative with multiple variables

Question

Let z = f(x,y), x = x(t,s) and y = y(t,s) all be twice continously differentiable functions

Try to find $$\frac{\partial z^2}{\partial t^2}$$

I've tried it and only got: $$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$$

Now the next step is

$$\frac{\partial }{\partial t}\frac{\partial z}{\partial t}$$

Anyone knows how to use the product and chain rule correctly here?

Thanks in advance!

Answer 1

$\frac{\partial^2z}{\partial t^2}=\frac{\partial}{\partial t}\left(\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}\right)+\frac{\partial}{\partial t}\left(\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}\right)$

To calculate $\frac{\partial}{\partial t}\left(\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}\right)$

$$ \begin{align}\frac{\partial}{\partial t}\left(\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}\right)&=\frac{\partial x}{\partial t}\frac{\partial}{\partial t}\left(\frac{\partial z}{\partial x}\right)+\frac{\partial z}{\partial x}\frac{\partial}{\partial t}\left(\frac{\partial x}{\partial t}\right) \text{by the product rule}\\&=\frac{\partial x}{\partial t}\frac{\partial^2z}{\partial x^2}\frac{\partial x}{\partial t}+ \frac{\partial x}{\partial t}\frac{\partial^2z}{\partial x\partial y}\frac{\partial y}{\partial t} +\frac{\partial z}{\partial x}\frac{\partial^2x}{\partial t^2} \text{ by the chain rule}\\&=\left(\frac{\partial x}{\partial t}\right)^2\frac{\partial^2z}{\partial x^2}+\frac{\partial x}{\partial t}\frac{\partial^2z}{\partial x\partial y}\frac{\partial y}{\partial t}+\frac{\partial z}{\partial x}\frac{\partial^2x}{\partial t^2}\end{align}$$

Answer 2

The procedure is systematic considering the (nested) dependence, e.g. $\dfrac{\partial z}{\partial x}$ is still some function of $x$ and $y$, both functions of $s$ and $t$ that yields $\dfrac{\partial }{\partial t}\dfrac{\partial z}{\partial x}=\dfrac{\partial^2 z}{\partial x^2}\dfrac{\partial x}{\partial t}+\dfrac{\partial^2 z}{\partial x\partial y}\dfrac{\partial y}{\partial t}$

$\dfrac{\partial }{\partial t}\dfrac{\partial z}{\partial t}=\dfrac{\partial^2z}{\partial t^2}=\dfrac{\partial}{\partial t}\left(\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial t}\right)+\dfrac{\partial}{\partial t}\left(\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial t}\right)$

Its parts:

$\dfrac{\partial}{\partial t}\left(\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial t}\right)=\dfrac{\partial }{\partial t}\left(\dfrac{\partial z}{\partial x}\right)\dfrac{\partial x}{\partial t}+\dfrac{\partial z}{\partial x}\dfrac{\partial^2 x}{\partial t^2}$

$\dfrac{\partial}{\partial t}\left(\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial t}\right)=\dfrac{\partial }{\partial t}\left(\dfrac{\partial z}{\partial y}\right)\dfrac{\partial y}{\partial t}+\dfrac{\partial z}{\partial y}\dfrac{\partial^2 y}{\partial t^2}$

Still:

$\dfrac{\partial}{\partial t}\left(\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial t}\right)=\dfrac{\partial^2 z}{\partial x^2}\left(\dfrac{\partial x}{\partial t}\right)^2+\dfrac{\partial^2 z}{\partial x\partial y}\dfrac{\partial y}{\partial t}\dfrac{\partial x}{\partial t}+\dfrac{\partial z}{\partial x}\dfrac{\partial^2 x}{\partial t^2}$

$\dfrac{\partial}{\partial t}\left(\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial t}\right)=\dfrac{\partial^2 z}{\partial y\partial x}\dfrac{\partial x}{\partial t}\dfrac{\partial y}{\partial t}+\dfrac{\partial^2 z}{\partial y^2}\left(\dfrac{\partial y}{\partial t}\right)^2+\dfrac{\partial z}{\partial y}\dfrac{\partial^2 y}{\partial t^2}$

And sum them.