# Derivative

**differentiationdifferentiablefirst derivativedifferentiatedrate of changedifferentiatingdifferentiateordinary derivativederivativesdifferential**

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).wikipedia

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### Time

**temporaldurationsequence of events**

For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

Time is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience.

### Tangent

**tangent linetangentialtangents**

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.

More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point (c, f(c)) on the curve and has slope f(c), where f is the derivative of f.

### Partial derivative

**partial derivativespartial differentiationpartial differential**

It can be calculated in terms of the partial derivatives with respect to the independent variables.

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

### Integral

**integrationintegral calculusdefinite integral**

The fundamental theorem of calculus relates antidifferentiation with integration.

Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other.

### Fundamental theorem of calculus

**First Fundamental Theorem Of Calculusfundamental theorem of real calculusfundamental theorem of the calculus**

The fundamental theorem of calculus relates antidifferentiation with integration.

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.

### Antiderivative

**indefinite integralindefinite integrationantidifferentiation**

The reverse process is called antidifferentiation.

whose derivative is equal to the original function

### Calculus

**infinitesimal calculusdifferential and integral calculusclassical calculus**

Derivatives are a fundamental tool of calculus.

However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".

### Matrix (mathematics)

**matrixmatricesmatrix theory**

The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables.

Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.

### Difference quotient

**Fermat's difference quotientchangedifference quotients**

The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences

which when taken to the limit as h approaches 0 gives the derivative of the function f.

### Limit of a function

**limitlimitsconvergence**

The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences

It also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.

### Infinitesimal

**infinitesimalsinfinitely closeinfinitesimally**

In Leibniz's notation, an infinitesimal change in

To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative).

### Leibniz's notation

**Leibniz notationby LeibnizLeibniz**

In Leibniz's notation, an infinitesimal change in

. If this is the case, then the derivative of

### Velocity

**velocitiesvelocity vectorlinear velocity**

For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

In terms of a displacement-time (x vs. t) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point, and the average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity.

### Function of several real variables

**functions of several real variablesReal multivariable functionmulti-variable function**

Derivatives may be generalized to functions of several real variables.

Elementary calculus is the calculus of real-valued functions of one real variable, and the principal ideas of differentiation and integration of such functions can be extended to functions of more than one real variable; this extension is multivariable calculus.

### Jacobian matrix and determinant

**Jacobian matrixJacobianJacobian determinant**

The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables.

is differentiable at a point

### Delta (letter)

**deltaΔdelta symbol**

(Delta) is an abbreviation for "change in", and the combinations \Delta x and \Delta y refer to corresponding changes, i.e.:.

### Limit (mathematics)

**limitlimitsconverge**

The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers.

Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

### Differentiable function

**differentiablecontinuously differentiabledifferentiability**

is said to be differentiable at

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

### Constant function

**constantconstant mapconstant mapping**

The simplest case, apart from the trivial case of a constant function, is when

In the context where it is defined, the derivative of a function is a measure of the rate of change of function values with respect to change in input values.

### History of calculus

**developmentdevelopment of calculusearlier mathematicians**

Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points.

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series.

### Linearity

**linearlinearlycomplex linear**

gives the best linear approximation

Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian.

### Second derivative

**concavityconcaveconcavities**

and is called the ''second derivative of

is the derivative of the derivative of

### Jerk (physics)

**jerkjoltJolt (physics)**

is the jerk.

In physics, jerk or jolt is the rate of change of acceleration; that is, the time derivative of acceleration, or the second derivative of velocity, or the third derivative of position.

### Differentiation rules

**Table of derivativesSum rule in differentiationsum rule**

By standard differentiation rules, if a polynomial of degree

This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

### Operator (mathematics)

**operatoroperatorsmathematical operators**

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions.

For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.