AP® Calculus BC · Unit 5 · Topic 5.12
Exploring Behaviors of Implicit Relations
On curves defined implicitly, horizontal tangents occur where dy/dx = 0 and vertical tangents occur where dy/dx is undefined. You find both by differentiating implicitly and checking numerator and denominator separately.
Implicit curves like x2 + xy + y2 = 3 aren't functions you can solve for y and then differentiate directly. Instead you differentiate both sides with respect to x, treating y as a hidden function of x, and solve for dy/dx. The result is almost always a fraction: some expression in x and y on top, some expression in x and y on bottom. That fraction is the key to the whole shape of the curve. Where the top equals zero (and the bottom doesn't), the tangent line is horizontal. Where the bottom equals zero (and the top doesn't), the tangent line is vertical — the slope blows up because the curve is momentarily standing straight up, not because anything breaks.
Mechanically, you set the numerator to zero and solve that equation together with the original curve equation, as a system. For x2 + xy + y2 = 3, implicit differentiation gives dy/dx = -(2x+y)/(x+2y). Setting 2x+y = 0 gives y = -2x; substituting into the original equation collapses it to 3x2 = 3, so x = 1 or x = -1, giving horizontal tangents at (1,-2) and (-1,2). Vertical tangents come from the same derivative: set x+2y = 0 instead, and solve alongside the curve equation.
This topic sits right after locating horizontal and vertical tangents on ordinary functions, and feeds straight into optimization and curve sketching once relations replace functions. The same logic — numerator tells you flat spots, denominator tells you steep spots — also extends to second derivatives, which you get by differentiating dy/dx implicitly again, useful for checking concavity on loops and ellipses where no single formula for y exists.
Key terms
- implicit differentiation
- Differentiating both sides of an equation in x and y, treating y as an unknown function of x and applying the chain rule to y-terms.
- horizontal tangent
- A point where dy/dx equals zero, found by setting the numerator of the implicit derivative to zero.
- vertical tangent
- A point where dy/dx is undefined because the denominator of the implicit derivative is zero, while the curve stays smooth.
- implicit relation
- An equation in x and y, like x2+xy+y2=3, that defines a curve without expressing y as a single function of x.
Watch it explained
- How it worksHow does an implicit derivative locate horizontal and vertical tangents on a closed loop?
- Worked exampleFind every point on x2 + xy + y2 = 3 where the tangent line is horizontal.
- Common mistakeWhy isn't a vertical tangent the same thing as a gap in the curve?
- Going furtherWhat happens to the second derivative of an implicitly defined curve?
Check yourself
End of topic.