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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''', one of the fundamental concepts of [[calculus]], measures how quickly a [[function]] changes as its input value changes. Given a graph of a [[real]] curve, the derivative at a specific point will equal the [[slope]] of the line [[tangent]] to that point. For example, the derivative of <tt>y = x²</tt> at the point <tt>(1,1)</tt> tells how quickly the function is increasing at that point.  If a function has a derivative at some point, it is said to be '''differentiable''' there.  If a function has a derivative at every point where it is defined, we say it is a '''differentiable function'''. 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).

 

One of the main applications of differential calculus is '''differentiating''' a function, or calculating its derivative. The First [[Fundamental Theorem of Calculus]] explains that one can find the original function, given its derivative, by [[integration|integrating]], or taking the integral of, the derivative.

 

The derivative of the function <tt>f(x)</tt>, <tt>f'(x)</tt> or :<math>frac{df}{dx}</math>, is defined as:

 

:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>

 

In other words, it is the limit of the [[slope]] of the [[secant]] line to f(x) as it becomes a [[tangent line]].  If the tangent line is increasing (which it is if the original function is increasing), the derivative is positive; if the function is decreasing, the derivative is negative.

 

In general, <tt>f'(mx) = m</tt>; that is, the derivative of any line is equal to its slope.

 

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.

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.

==Useful rules for finding derivatives==

==Rules for finding derivatives==

*[[Power rule]]

*[[Power rule]]

*[[Constant-multiple rule]]

*[[Constant-multiple rule]]