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  "body": "\n## Summary\n\nImplicit differentiation finds the derivative of a function that an equation $F(x, y) = 0$ defines, without solving the equation for $y$.\nOne differentiates both sides of $F(x, y(x)) = 0$ with the [[chain rule|0EAAS2EHDR]] and solves the result for $y'(x)$.\n\n## Remarks\n\nIf $F$ is of class $C^1$ and $\\partial F / \\partial y \\ne 0$ at the point, then $y'(x) = -\\dfrac{\\partial F / \\partial x}{\\partial F / \\partial y}$, evaluated at $(x, y(x))$. The [[implicit function theorem|0BXBADR94C]] guarantees that $y(x)$ exists and is differentiable.\nFor the circle $x^2 + y^2 = 1$, differentiation gives $2 x + 2 y y' = 0$. So $y' = -x / y$ at each point with $y \\ne 0$, on both the upper and the lower arc.\nWith several variables, $F(x, y, z) = 0$ gives $\\partial z / \\partial x = -F_x / F_z$ and $\\partial z / \\partial y = -F_y / F_z$ where $F_z \\ne 0$. Here $F_x$, $F_y$, $F_z$ are the [[partial derivatives|05AXG3QRKR]] of $F$.\n\n## Explanations\n\n### L1 human\n\nSometimes a curve is given by an equation and not by a formula $y = f(x)$.\nYou can still find its slope: treat $y$ as a function of $x$, differentiate the equation term by term, and solve for $y'$.\n",
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