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suppose i have a function ##f( x , y)##. i make a change of variables such that ##z(x,y)## in such a way that now ##f( z , y)##. how do i find $$\frac{\partial f}{\partial y}$$ $$\frac{\partial f}{\partial x}$$ $$\frac{\partial^2 f}{\partial y^2}$$ $$\frac{\partial^2 f}{\partial x}$$

i think $$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial z}\frac{\partial z}{\partial x}$$ and $$\frac{\partial^2 f}{\partial x^2} = \frac{\partial^2 f}{\partial z^2} \frac{\partial z}{\partial x} + \frac{\partial f}{\partial z}\frac{\partial^2 z}{\partial x^2}$$

i also think $$\frac{\partial f}{\partial y} = \frac{\partial f}{\partial z} \frac{\partial z}{\partial y} + \frac{\partial f}{\partial y}$$ but something is wrong here. i feel that i need some new notation or something to fully represent what is happening.

i have no idea how to express $$\frac{\partial^2 f}{\partial y^2}$$

any help is greatly appreciated

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# Multidimensional chain rule (tough)

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