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Looking to verify whether the following formulation is correct. Suppose we have the following function, relationships:

$$y=f(x)$$ $$x=g(a,b)$$ $$y=f[g(a,b)]$$

Is the below correct (including notation)? $$\frac{dy}{dx}=\frac{\partial y}{\partial a}+\frac{\partial y}{\partial b}$$ $$\frac{dy}{dx}=\frac{\partial y}{\partial x}\frac{\partial x}{\partial a}+\frac{\partial y}{\partial x}\frac{\partial x}{\partial b}$$

In words, the total derivative of the composite function $y$ with respect to $x$ is the sum of the partial derivatives of $y$ with respect to $a$ and $b$

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No, this is in general not true. For example, consider \begin{align*} y(x)&:=x,\\ x(a,b):&=a+b. \end{align*} Then we have $$\frac{\partial y}{\partial x}=1$$ but $$\frac{\partial y}{\partial a} + \frac{\partial y}{\partial b}=2. $$

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