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A '''derivative''' is a measure in [[calculus]] of how functions change based on how their input values change. Given a graph of a [[real]] curve, the derivative at a specific point will equal the [[slope]] of the line [[tangent]] to that point. If a function has a derivative at some point, it is said to be '''differentiable''' there, and in general we call a function differentiable whenever it has a derivative at every point at which it is defined. Differentiability implies [[continuous|continuity]]  as well as [[integral|integrability]] on bounded [[subset]]s of the [[domain]].

A '''derivative''' is a measure in [[calculus]] of how functions change based on how their input values change. Given a graph of a [[real]] curve, the derivative at a specific point will equal the [[slope]] of the line [[tangent]] to that point. If a function has a derivative at some point, it is said to be '''differentiable''' there, and in general we call a function differentiable whenever it has a derivative at every point at which it is defined. Differentiability implies [[continuous|continuity]]  as well as [[integral|integrability]] on bounded [[subset]]s of the [[domain]].

To calculate the derivative of a function, one must use techniques from the differential branch of calculus. This branch of calculus is related to the integral branch by the first [[Fundamental Theorem of Calculus]]: ''[[differentiation]]'' (the process of finding a derivative) ''is the reverse process of [[integration]]'' (the process of finding an integral).

To calculate the derivative of a function, one must use techniques from the differential branch of calculus. This branch of calculus is related to the integral branch by the first [[Fundamental Theorem of Calculus]]: ''differentiation'' (the process of finding a derivative) ''is the reverse process of [[integration]]'' (the process of finding an integral).

In mathematics, derivatives are helpful in determining the [[maxima|maximum]] and [[minima|minimum]] of a function. For example, taking the derivative of a [[quadratic]] function will yield a linear function. The points at which this function equals zero are called [[critical point]]s. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the ''second derivative test''. The second derivative is used to determine the [[concavity]], or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of ''inflection,'' meaning that it is a point where the concavity could be changing.

In mathematics, derivatives are helpful in determining the [[maxima|maximum]] and [[minima|minimum]] of a function. For example, taking the derivative of a [[quadratic]] function will yield a linear function. The points at which this function equals zero are called [[critical point]]s. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the ''second derivative test''. The second derivative is used to determine the [[concavity]], or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of ''inflection,'' meaning that it is a point where the concavity could be changing.