Implicit Differentiation
Implicit differentiation finds dy/dx for equations where y is not isolated, by differentiating both sides with respect to x.
Formula
\frac{d}{dx}[F(x,y)] = 0, \text{ solve for } \frac{dy}{dx}
Definition
Sometimes an equation mixes $x$ and $y$ together and you cannot easily solve for $y$; implicit differentiation lets you find the slope $dy/dx$ anyway by differentiating both sides of $F(x, y) = 0$ with respect to $x$, treating $y$ as a function of $x$ and applying the chain rule whenever $y$ appears, then solving algebraically for $dy/dx$. By the Implicit Function Theorem, if $F(x, y) = 0$ and $F_y \neq 0$ at a point, then $y$ is locally a differentiable function of $x$ and $dy/dx = -F_x/F_y$, which generalizes to multivariable settings via the Jacobian.
Example
The equation $x^2 + y^2 = 25$ describes a circle; differentiating both sides gives $2x + 2y(dy/dx) = 0$, so $dy/dx = -x/y$, matching what the Implicit Function Theorem predicts ($F_x = 2x$, $F_y = 2y$). For the folium of Descartes, $x^3 + y^3 = 6xy$: $3x^2 + 3y^2(dy/dx) = 6y + 6x(dy/dx)$, rearranging to $dy/dx = (2y - x^2)/(y^2 - 2x)$.
Key Insight
Implicit differentiation is like finding the slope of a curved wall without having to tear it apart first, and it is essential for curves defined by equations that cannot be written as $y = f(x)$, including circles, ellipses, and many algebraic curves. The Implicit Function Theorem is a cornerstone of differential geometry and manifold theory, guaranteeing local chart representations for smooth hypersurfaces.