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Line 1: − + − Essentially, differentiation is employed as a means to calculate the [[gradient]] or rate of change at a particular value for a given function, ''f''. Consequently, it can be used to calculate velocity from a displacement-time graph, or acceleration from a velocity-time graph. Furthermore, it can be used to calculate the rate of cooling from a temperature-time graph. These are a few examples of the applications of differentiation.+ + + + + Line 47:
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In mathematics a '''derivative''' is measure of how functions change. Algebraic '''differentiation''' is an important part of [[calculus]], an essential branch of [[mathematics]]. Differentiation can be used, for example, in [[mechanics]] to find the acceleration of an object from a velocity-time graph.
In mathematics a '''derivative''' is the change in the function with respect to one of it vaiables. Essentially, the derivative is a means to calculate the [[gradient]] or rate of change at a particular value for a given function, ''f''. Consequently, it can be used to calculate velocity from a displacement-time graph, or acceleration from a velocity-time graph. Furthermore, it can be used to calculate the rate of cooling from a temperature-time graph.
The process of finding a derivative is called '''differentiation'''.
Algebraic differentiation is an important part of [[calculus]], an essential branch of [[mathematics]].
For a single variable real function the derivative is the equation that given the [[gradient (two points)|slope]] of the line which is tangential at that point.
When defined from the first principals, the derivative of a function is the limit of the average rate of change of the function over <math>[x,x+h]</math> as <math>h</math> tends to zero. In other words, the derivative
When defined from the first principals, the derivative of a function is the limit of the average rate of change of the function over <math>[x,x+h]</math> as <math>h</math> tends to zero. In other words, the derivative
The roots of differentiation are profoundly linked with tangency; ergo, this aspect of mathematics can first be perceived to have been developed during the time of the [[Ancient Greeks]] through the work of Greek geometers like [[Euclid]], [[Sanath]] and [[Archimides]].
The roots of differentiation are profoundly linked with tangency; ergo, this aspect of mathematics can first be perceived to have been developed during the time of the [[Ancient Greeks]] through the work of Greek geometers like [[Euclid]], [[Sanath]] and [[Archimides]].
==Partial derivatives==
For multivalue functions <math>f(x_{1},x_{2},\dots,x_{i},\dots,x_{n})</math>, the partial derivative with respect to <math>x_{i}</math> is defined as,
:<math>\frac{\partial f(x_{1},x_{2},\dots,x_{i},\dots,x_{n})}{\partial x_{i}}=\lim_{h\rightarrow0}\frac{f(x_{1},x_{2},\dots,x_{i}+h,\dots,x_{n})-f(x_{1},x_{2},\dots,x_{i},\dots,x_{n})}{h}</math>
[[Category:Calculus]]
[[Category:Calculus]]
[[Category: Mathematics]]
[[Category: Mathematics]]