AP® Calculus BC · Unit 3 · Topic 3.2
Implicit Differentiation
Implicit differentiation finds dy/dx for equations where y isn't solved for explicitly, by differentiating both sides with respect to x and treating y as a hidden function of x, which brings in a dy/dx factor every time you differentiate a y term.
A circle like x2 + y2 = 25 isn't a function of x — for most x values there are two y's. But zoom in on any single point of the circle and the curve looks like a straight line with one definite slope. That slope is what implicit differentiation finds, without ever solving for y.
The trick is to treat y as some unknown function of x, even though you can't write down its formula. Then you differentiate both sides of the equation with respect to x, term by term. Any plain x term differentiates as usual. Any y term needs the chain rule, because y is really y(x): differentiating y2 gives 2y times dy/dx, not just 2y, since you're differentiating a function of a function. Once every term is differentiated, dy/dx appears algebraically somewhere in the equation, and you solve for it like any variable.
This is the same chain-rule logic that governs composite functions, just applied term by term inside an equation instead of to one formula. It also sets up inverse function differentiation next, where you differentiate an equation like x = sin(y) implicitly to find dy/dx for y = arcsin(x). And it explains vertical tangents: wherever the implicit derivative has a zero denominator, the curve's slope becomes undefined, meaning the tangent line is vertical there.
Key terms
- Implicit function
- A relationship between x and y where y is not isolated on one side, so it isn't written as an explicit formula in x.
- Implicit differentiation
- Differentiating both sides of an equation with respect to x while treating y as an unknown function of x.
- dy/dx factor
- The extra derivative term the chain rule inserts whenever you differentiate an expression containing y with respect to x.
- Vertical tangent
- A point on a curve where the slope is undefined because the implicit derivative has a zero denominator.
Watch it explained
- How it worksHow does treating y as a hidden function of x let me differentiate both sides?
- Worked exampleFind dy/dx for x2 + y2 = 25 and evaluate it at the point (3, 4).
- Worked exampleFind dy/dx for x3 + y3 = 6xy at the point (3, 3).
- Common mistakeWhy doesn't differentiating y2 with respect to x simply give 2y?
- Going furtherWhat happens to the implicit slope where the curve turns vertical?
Check yourself
End of topic.