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In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: [ 11 ] Linearity : For constants a and b and differentiable functions f and g , d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.}
Differential quadrature is the approximation of derivatives by using weighted sums of function values. [22] [23] Differential quadrature is of practical interest because its allows one to compute derivatives from noisy data.
Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena . Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis , functional analysis , differential geometry ...
Compute analytic derivatives of any Fortran77, Fortran95, or C program through a web-based interface Automatic Differentiation of Fortran programs; Description and example code for forward Automatic Differentiation in Scala Archived 2016-08-03 at the Wayback Machine
GeoGebra can store variables for numbers, vectors and points, calculate derivatives and integrals of functions, and has a full complement of commands like Root or Extremum. Teachers and students can use GeoGebra as an aid in formulating and proving geometric conjectures. GeoGebra's main features are: Interactive geometry environment (2D and 3D)
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}
For this reason, the derivative is sometimes called the slope of the function f. [48]: 61–63 Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2 be the squaring function. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is ...