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Suppose $z = x + y$ and in addition, there is a function $f(z)=F(x,y)$ for all values. Is there a way to relate,

$$\frac{df}{dz}, \frac{\partial F}{\partial x}\text{ and }\frac{\partial F}{\partial y}\textrm{?}$$

In general, suppose $z = g(x,y)$, what is the relationship (if any)?

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1 Answers 1

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We're given $F(x,y)=f(x+y)$. So $\frac{\partial F}{\partial x} (x,y)= f'(x+y)\cdot 1=\frac{\partial F}{\partial y}(x,y)$, by the Chain Rule.

In general, if $F(x,y)=f(g(x,y))$, we only have the statement of the Chain Rule. $\frac{\partial F}{\partial x}(x,y) = f'(g(x,y)) \frac{\partial g}{\partial x} (x,y)$ and $\frac{\partial F}{\partial y}(x,y) = f'(g(x,y)) \frac{\partial g}{\partial y} (x,y)$.