Implicit Function Theorem

When F(x,y)=0 defines y as a function of x, and what its slope is.

The idea

Theorem (Implicit function theorem).

Let $F$ be $C^{1}$ on an open set containing $(a,b)$, and suppose $F(a,b) = 0 \qquad \text{and} \qquad \frac{\partial F}{\partial y}(a,b) \ne 0.$ Then there is an open interval around $a$ on which a unique $C^{1}$ function $g$ is defined with $g(a) = b$ and $F\bigl(x, g(x)\bigr) = 0$, and its derivative is $g'(x) = -\frac{\partial F/\partial x}{\partial F/\partial y},$ the partials being evaluated at $\bigl(x, g(x)\bigr)$.

The formula comes from the chain rule. Differentiating the identity $F(x, g(x)) = 0$ in $x$ gives $F_x + F_y\,g'(x) = 0$, and the hypothesis $F_y \ne 0$ is precisely what allows solving this for $g'$. The theorem thereby delivers $g'$ without ever producing $g$: no formula for $y$ in terms of $x$ is needed, and for most equations none exists.

The conclusion is local. It describes the solution set only near $(a,b)$ and claims nothing about the curve elsewhere: a curve can cross itself, or double back, and still be the graph of a function near each point where the hypotheses hold. At a point where $F_y = 0$, the theorem makes no claim.

Ways to work on it

• Walkthrough. When an equation defines a function implicitly, and how to compute its slope.
• Practice. Compute the slope of an implicitly defined curve at a point.
• Hardest. Find slopes on an implicit cubic curve, including where the theorem fails.

Not sure where to start? Take the ten-question placement test.