Derivative

The derivative of a function measures its instantaneous rate of change with respect to its input. Geometrically, it is the slope of the function’s tangent line at a point.

Limit (mathematics): A value that a function or sequence approaches as its input approaches a specified point or grows without bound. The derivative is defined by a limit of average rates of change.
Function (mathematics): A relation that assigns exactly one output to each input in its domain. A derivative describes how a function’s output responds to changes in its input.
Velocity: The rate of change of position with respect to time, including direction. For a position function, its derivative with respect to time is velocity.
Integral: A mathematical quantity that represents accumulation, signed area, or the limit of sums. Differentiation measures local change, while integration accumulates change across an interval.
Difference quotient: An expression comparing the change in a function’s value with a change in its input. Its limit as the input change shrinks gives the derivative.
Average rate of change: The change in a quantity divided by the corresponding change in its independent variable over an interval. Instantaneous rate emerges as the interval used for this rate shrinks.
Acceleration: The rate of change of velocity with respect to time. It is the second derivative of position with respect to time.
Finite difference: A change between function values at distinct inputs, often used to approximate a derivative numerically. It estimates local change without taking the exact limiting process in the derivative definition.
Tangent line: A line that locally matches a curve at a point and, for a differentiable curve, has the curve’s derivative as its slope. The derivative gives the tangent line’s slope at each point.
Continuity (mathematics): A property of a function whose values approach its value at a point as the input approaches that point. Differentiability at a point implies continuity there, but continuity alone does not guarantee a derivative.

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