Implicit Differentiation

Differentiate both sides; chain-rule every y; solve for dy/dx.

The idea

Implicit differentiation finds the slope $\frac{dy}{dx}$ along a curve that cannot be solved for $y$. A circle cannot be written as $y = f(x)$, since a vertical line meets it twice; a relation such as $x^{2} + xy + y^{2} = 7$ cannot be solved for $y$ by elementary means at all. Near most of its points the relation still determines $y$ as a function of $x$ — implicitly, without a formula.

To differentiate it, treat $y$ as an unnamed function of $x$ and differentiate both sides of the equation with respect to $x$. Terms built only from $x$ behave as usual. Any term containing $y$ is a composition — something applied to $y$, with $y$ itself a function of $x$ — so the chain rule applies and the term acquires a factor of $\frac{dy}{dx}$: $y^{2}$ differentiates to $2y \frac{dy}{dx}$, not to $2y$.

The result is an equation in which $\frac{dy}{dx}$ is an unknown. Collect the terms carrying it, factor it out, and divide.

The answer normally involves both $x$ and $y$: one value of $x$ can meet the curve at several points with different tangents, so the slope depends on the point, not on $x$ alone.

Ways to work on it

• Walkthrough. Differentiate a circle implicitly and find a tangent slope.
• Practice. Slope of xy = c at a point.
• Hardest. Implicit slope of a mixed quadratic relation.

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