$x,y$ are independent variables and $f = f(x,y)$. Some other variable $z = z(x,y)$. I want to calculate $\frac{df}{dz}$.
I started as follows, $$\frac{df}{dz} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial z} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial z} $$ Is it correct?
Suppose $f = x+y$ and also $z=x+y$, so $\frac{df}{dz} = 1$. On the other hand $\frac{\partial f}{\partial y} = 1=\frac{\partial f}{\partial x} $ and $\frac{\partial y}{\partial z} =1 = \frac{\partial x}{\partial z}$
so, $\frac{\partial f}{\partial x}\frac{\partial x}{\partial z} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial z} = 2$
can anyone help me figure out what went wrong?
What do you mean/hope to calculate by $\mathrm df/\mathrm dz$? It would be relatively common to write $\mathrm df=\dfrac{\partial f}{\partial x}\,\mathrm dx+\dfrac{\partial f}{\partial y}\,\mathrm dy$ and similarly for $z$ in place of $f$. But then if you tried to write $\mathrm df/\mathrm dz$, you'd usually get weird things that don't simplify (or even make sense?) like $\dfrac{5\,\mathrm dx+x\,\mathrm dy}{xy^2\,\mathrm dx+e^y\,\mathrm dy}$.