# Combined Partial Derivative?

$${\partial x \over \partial y_1} = f_1(y_1)$$

$${\partial x \over \partial y_2} = f_2(y_2)$$

$${\partial y_1 \over \partial z} = f_3(z)$$

$${\partial y_2 \over \partial z} = f_4(z)$$

Given the above, what is ${\partial x \over \partial z}$ equal to?

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Hint: Chain rule. –  Neal Nov 2 '12 at 4:17
I know how to apply chain rule with a 1D list, but how does it apply to this diamond shaped graph? –  Andrew Tomazos Nov 2 '12 at 4:19
Why can't you use $\frac{\partial x}{\partial z}=\frac{\partial x}{\partial y_1}\frac{\partial y_1}{\partial z}+\frac{\partial x}{\partial y_2}\frac{\partial y_2}{\partial z}$? Perhaps it would be clearer if you specified what $x,y_1,y_2,z$ are. This is why I hate $\frac{\partial f}{\partial x}$ notation (vs. $D_1 f$). –  wj32 Nov 2 '12 at 4:33