AP Calculus AB and BC glossary

Vertical tangent

A vertical tangent occurs at a point where the derivative grows without bound with the same sign from both sides, so the tangent line is vertical. The function remains continuous there but is not differentiable, since infinite slope is not a number.

The cube root function f(x)=x1/3f(x) = x^{1/3} has a vertical tangent at the origin. Its derivative 13x−2/3\frac{1}{3}x^{-2/3} tends to +∞+\infty from both sides, so the curve passes through smoothly while standing straight up.

Vertical tangents matter for implicit differentiation, where dydx\frac{dy}{dx} often has a denominator that vanishes. Setting that denominator to zero locates the points with vertical tangents.

Appears in: Unit 2: Defining the Derivative