Why there should be a negative sign in the chain rule?

Question

Suppose I have a function $$f(x,y)=0$$ and suppose I want to find $\frac{dx}{dy}$ then I think I can get it by doing $$\frac{dx}{dy}=\frac{df}{dy}\frac{dx}{df}.$$ But I do not understand why do we need to put a negative sign in front of $\frac{df}{dy}\frac{dx}{df}$. Please explain. Thanks in advance.

Answer 1

The exact rule is

$$\frac{\partial f(x,y)}{\partial x}dx+\frac{\partial f(x,y)}{\partial y}dy=0\implies\frac{dy}{dx}=-\dfrac{\dfrac{\partial f(x,y)}{\partial x}}{\dfrac{\partial f(x,y)}{\partial y}}.$$

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E.g., with $$x^2+y^2-1=0$$

$$\frac{dy}{dx}=-\frac{2x}{2y}.$$

Compare to

$$y=\sqrt{1-x^2}$$ then

$$\frac{dy}{dx}=-\frac{2x}{2\sqrt{1-x^2}}$$ (and similarly with the negative branch).

Answer 2

$f$ is a function of two variables. It does not make sense to write $\frac{df}{dy}$.

If $f(x,y)=0$, the derivative of the implicit function $x(y)$ is given by:

$${\displaystyle {\frac {dx}{dy}}=-{\frac {\partial f/\partial y}{\partial f/\partial x}}=-{\frac {f_{y}}{f_{x}}},} $$ where $f_x$ and $f_y$ indicate the partial derivatives of $f$ with respect to $x$ and $y$.

The above formula comes from using the generalized chain rule to obtain the total derivative—with respect to $y$—of both sides of $f(x, y) = 0$:

$${\displaystyle {\frac {\partial f}{\partial x}}{\frac {dx}{dy}}+{\frac {\partial f}{\partial y}}{\frac {dy}{dy}}=0,} $$ hence $$ {\displaystyle {\frac {\partial f}{\partial y}} +{\frac {\partial f}{\partial x}}{\frac {dx}{dy}}=0,} $$ which, when solved for $dx/dy$, gives the expression above.