Implicit differentiation

Summary

Implicit differentiation finds the derivative of a function that an equation $F(x, y) = 0$ defines, without solving the equation for $y$. One differentiates both sides of $F(x, y(x)) = 0$ with the chain rule and solves the result for $y'(x)$.

Remarks

If $F$ 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 guarantees that $y(x)$ exists and is differentiable. For 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. With 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 of $F$.

Explanations

L1 human

Sometimes a curve is given by an equation and not by a formula $y = f(x)$. You can still find its slope: treat $y$ as a function of $x$, differentiate the equation term by term, and solve for $y'$.