# Derivative vs. Integral — What's the Difference?

## Difference Between Derivative and Integral

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## Compare with Definitions

#### Derivative

In mathematics, 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). Derivatives are a fundamental tool of calculus.

Oct 19, 2021

#### Integral

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration.

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#### Derivative

Resulting from or employing derivation

A derivative word.

A derivative process.

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#### Integral

Necessary to make a whole complete; essential or fundamental

Games are an integral part of the school's curriculum

Systematic training should be integral to library management

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#### Derivative

Copied or adapted from others

A highly derivative prose style.

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#### Integral

Of or denoted by an integer.

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#### Derivative

Something derived.

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#### Integral

A function of which a given function is the derivative, i.e. which yields that function when differentiated, and which may express the area under the curve of a graph of the function.

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#### Derivative

(Linguistics) A word formed from another by derivation, such as electricity from electric.

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#### Integral

Essential or necessary for completeness; constituent

The kitchen is an integral part of a house.

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#### Derivative

The limiting value of the ratio of the change in a function to the corresponding change in its independent variable.

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#### Integral

Possessing everything essential; entire.

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#### Derivative

The instantaneous rate of change of a function with respect to its variable.

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#### Integral

Expressed or expressible as or in terms of integers.

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#### Derivative

The slope of the tangent line to the graph of a function at a given point. Also called differential coefficient, fluxion.

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#### Integral

Expressed as or involving integrals.

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#### Derivative

(Chemistry) A compound derived or obtained from another and containing essential elements of the parent substance.

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#### Integral

A complete unit; a whole.

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#### Derivative

A financial instrument that derives its value from another more fundamental asset, as a commitment to buy a bond for a certain sum on a certain date.

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#### Integral

A number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or areal measurement of that unit, and all such products are summed.

Oct 19, 2021

#### Derivative

Obtained by derivation; not radical, original, or fundamental.

A derivative conveyance

A derivative word

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#### Integral

A definite integral.

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#### Derivative

Imitative of the work of someone else.

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#### Integral

An indefinite integral.

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#### Derivative

Referring to a work, such as a translation or adaptation, based on another work that may be subject to copyright restrictions.

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#### Integral

Constituting a whole together with other parts or factors; not omittable or removable

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#### Derivative

(finance) Having a value that depends on an underlying asset of variable value.

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#### Integral

(mathematics) Of, pertaining to, or being an integer.

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#### Derivative

Lacking originality.

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#### Integral

(mathematics) Relating to integration.

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#### Derivative

Something derived.

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#### Integral

(obsolete) Whole; undamaged.

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#### Derivative

(linguistics) A word that derives from another one.

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#### Integral

(mathematics) One of the two fundamental operations of calculus (the other being differentiation), whereby a function's displacement, area, volume, or other qualities arising from the study of infinitesimal change are quantified, usually defined as a limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.

The integral of a univariate real-valued function is the area under its curve; but be warned! Not all functions are integrable!

Oct 19, 2021

#### Derivative

(finance) A financial instrument whose value depends on the valuation of an underlying asset; such as a warrant, an option etc.

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#### Integral

(specifically) Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.

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#### Derivative

(chemistry) A chemical derived from another.

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#### Integral

(mathematics) A definite integral: the result of the application of such an operation onto a function and a suitable subset of the function's domain: either a number or positive or negative infinity. In the former case, the integral is said to be finite or to converge; in the latter, the integral is said to diverge. In notation, the domain of integration is indicated either below the sign, or, if it is an interval, with its endpoints as sub- and super-scripts, and the function being integrated forming part of the integrand (or, generally, differential form) appearing in front of the integral sign.

The integral of $\backslash frac\{1\}\{x\}$ on $[\backslash frac\{1\}\{2\},\; 1]$ is $\backslash ln(2)$, but the integral of the same function on $(0,\; 1]$ diverges. In notation, $\backslash int\_\backslash frac\{1\}\{2\}^1\backslash frac\{1\}\{x\}\; dx\; =\; \backslash ln(2)$, but $\backslash int\_0^1\backslash frac\{1\}\{x\}\; dx\; =\; \backslash infty$.

Oct 19, 2021

#### Derivative

(calculus) One of the two fundamental objects of study in calculus (the other being integration), which quantifies the rate of change, tangency, and other qualities arising from the local behavior of a function.

Oct 19, 2021

#### Integral

(mathematics) An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;

The integral of $x^2$ is $\backslash frac\{x^3\}\{3\}$ plus a constant.

Oct 19, 2021

#### Derivative

The derived function of $f(x)$: the function giving the instantaneous rate of change of $f$; equivalently, the function giving the slope of the line tangent to the graph of $f$. Written $f\text{'}(x)$ or $\backslash frac\{df\}\{dx\}$ in Leibniz's notation, $\backslash dot\{f\}(x)$ in Newton's notation (the latter used particularly when the independent variable is time). Category:en:Functions

The derivative of $x^2$ is $2x$; if $f(x)\; =\; x^2$, then $f\text{'}(x)\; =\; 2x$

Oct 19, 2021

#### Integral

The fluent of a given fluxion in Newtonian calculus.

Oct 19, 2021

#### Derivative

The value of such a derived function for a given value of its independent variable: the rate of change of a function at a point in its domain.

The derivative of $f(x)=x^3$ at $x=2$ is 12.

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#### Integral

Lacking nothing of completeness; complete; perfect; uninjured; whole; entire.

A local motion keepeth bodies integral.

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#### Derivative

(Of more general classes of functions) Any of several related generalizations of the derivative: the directional derivative, partial derivative, Fréchet derivative, functional derivative, etc.

Oct 19, 2021

#### Integral

Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.

Ceasing to do evil, and doing good, are the two great integral parts that complete this duty.

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#### Derivative

(generally) The linear operator that maps functions to their derived functions, usually written $D$; the simplest differential operator.

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#### Integral

Of, pertaining to, or being, a whole number or undivided quantity; not fractional.

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#### Derivative

Obtained by derivation; derived; not radical, original, or fundamental; originating, deduced, or formed from something else; secondary; as, a derivative conveyance; a derivative word.

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#### Integral

A whole; an entire thing; a whole number; an individual.

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#### Derivative

Hence, unoriginal (said of art or other intellectual products.

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#### Integral

An expression which, being differentiated, will produce a given differential. See differential Differential, and Integration. Cf. Fluent.

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#### Derivative

That which is derived; anything obtained or deduced from another.

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#### Integral

The result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)

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#### Derivative

A word formed from another word, by a prefix or suffix, an internal modification, or some other change; a word which takes its origin from a root.

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#### Integral

Existing as an essential constituent or characteristic;

The Ptolemaic system with its built-in concept of periodicity

A constitutional inability to tell the truth

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#### Derivative

A chord, not fundamental, but obtained from another by inversion; or, vice versa, a ground tone or root implied in its harmonics in an actual chord.

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#### Integral

Constituting the undiminished entirety; lacking nothing essential especially not damaged;

A local motion keepeth bodies integral

Was able to keep the collection entire during his lifetime

Fought to keep the union intact

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#### Derivative

An agent which is adapted to produce a derivation (in the medical sense).

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#### Derivative

A derived function; a function obtained from a given function by a certain algebraic process.

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#### Derivative

A substance so related to another substance by modification or partial substitution as to be regarded as derived from it; thus, the amido compounds are derivatives of ammonia, and the hydrocarbons are derivatives of methane, benzene, etc.

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#### Derivative

The result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx

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#### Derivative

A financial instrument whose value is based on another security

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#### Derivative

(linguistics) a word that is derived from another word;

`electricity' is a derivative of `electric'

Oct 19, 2021

#### Derivative

Resulting from or employing derivation;

A derivative process

A highly derivative prose style

Oct 19, 2021

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