A tangent vector is a mathematical concept used in calculus and geometry to describe the direction and rate of change of a curve at a specific point. It represents a vector that is tangent to the curve, meaning it touches the curve at that point without crossing it. Tangent vectors are essential in understanding how curves behave and are often used in fields like physics and engineering. In the context of a curve defined by a function, the tangent vector can be calculated using the derivative of the function at that point. This derivative gives the slope of the curve, which indicates how steeply it rises or falls. Tangent vectors are also fundamental in the study of manifolds and differential geometry, where they help describe the local properties of curved spaces.