Difference between revisions of "Derivative (calculus)"

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Derivatives are also useful in [[physics]], under the "rate of change" concept. For example, [[acceleration]] is the derivative of [[velocity]] with respect to time, and velocity is the derivative of [[distance]] with respect to time.
 
Derivatives are also useful in [[physics]], under the "rate of change" concept. For example, [[acceleration]] is the derivative of [[velocity]] with respect to time, and velocity is the derivative of [[distance]] with respect to time.
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==Useful rules for finding derivatives==
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[[Power rule]]
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[[Constant-multiple rule]]
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[[sum rule]]
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[[Chain rule]]
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[[Product rule]]
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[[Quotient rule]]
  
 
[[Category:Calculus]]
 
[[Category:Calculus]]
 
[[Category:Differentiation]]
 
[[Category:Differentiation]]

Revision as of 02:19, August 24, 2009

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 continuity as well as integrability on bounded subsets 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).

In mathematics, derivatives are helpful in determining the maximum and 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 points. 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.

Derivatives are also useful in physics, under the "rate of change" concept. For example, acceleration is the derivative of velocity with respect to time, and velocity is the derivative of distance with respect to time.

Useful rules for finding derivatives

Power rule Constant-multiple rule sum rule Chain rule Product rule Quotient rule