Derivative vs. Integral: Know the Difference

By Shumaila Saeed || Updated on December 26, 2023

Derivative measures the rate of change of a function, while an integral computes the area under the curve of a function.

Derivative vs. Integral

Key Differences

A derivative, in calculus, is a measure of how a function changes as its input changes, essentially representing the rate of change or slope of the function at a given point. An integral, on the other hand, is a mathematical concept that represents the accumulation of quantities, often visualized as the area under a curve of a function over an interval. Both concepts are fundamental in calculus, but they serve distinct purposes.

Dec 26, 2023

The derivative of a function at a point gives the instantaneous rate of change of the function at that point, akin to finding the velocity of an object in motion at a specific moment. Conversely, the integral of a function over an interval calculates the total accumulation, similar to determining the total distance traveled by an object over a period. While derivatives break functions down to understand instantaneous changes, integrals accumulate these changes over intervals.

Dec 26, 2023

Derivatives are used to find local behaviors of functions, like maxima, minima, and points of inflection, which are crucial in fields like physics and engineering. Integrals, in contrast, are used to calculate quantities like area, volume, and total growth, playing a vital role in applications ranging from area calculation to probability. Both are tools for understanding different aspects of functions and their behaviors.

Dec 26, 2023

In practical terms, when a scientist calculates the derivative of a position function, they obtain velocity, showcasing how fast the position is changing at any moment. When they compute the integral of a velocity function, they get the total change in position, or displacement, over a time interval. Derivatives and integrals, though inverse operations, are both necessary for a comprehensive understanding of dynamic systems.

Dec 26, 2023

Comparison Chart

Function

Measures rate of change of a function.

Calculates the area under the curve of a function.

Dec 26, 2023

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Mathematical Operation

Differentiation.

Integration.

Dec 26, 2023

Application

Used for finding slopes, velocities, and rates.

Used for finding areas, volumes, and totals.

Dec 26, 2023

Concept

Analyzes local behavior of functions.

Accumulates values over an interval.

Dec 26, 2023

Derivative and Integral Definitions

Derivative

In calculus, the rate of change of a function.

The derivative of the velocity function gives us acceleration.

Dec 26, 2023

Integral

In mathematics, the antiderivative or integral of a function.

Calculating the integral of the function helped find the area under the curve.

Dec 26, 2023

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Derivative

Imitative of the work of another person.

The film was dismissed as a derivative of classic Hollywood horror.

Dec 26, 2023

Integral

Complete and necessary for completeness.

Participation is an integral aspect of the learning process.

Dec 26, 2023

Derivative

Resulting from or employing derivation

A derivative word.

A derivative process.

Oct 19, 2023

Integral

Constituting a whole together with other parts.

The new system is an integral component of the software architecture.

Dec 26, 2023

Derivative

Copied or adapted from others

A highly derivative prose style.

Oct 19, 2023

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Integral

Composed of parts that together constitute a whole.

Each department plays an integral role in the company's success.

Dec 26, 2023

Derivative

Something derived.

Oct 19, 2023

Integral

Essential or necessary for completeness; constituent

The kitchen is an integral part of a house.

Oct 19, 2023

Derivative

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

Oct 19, 2023

Integral

Possessing everything essential; entire.

Oct 19, 2023

Derivative

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

Oct 19, 2023

Integral

Expressed or expressible as or in terms of integers.

Oct 19, 2023

Derivative

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

Oct 19, 2023

Integral

Expressed as or involving integrals.

Oct 19, 2023

Derivative

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

Oct 19, 2023

Integral

A complete unit; a whole.

Oct 19, 2023

Derivative

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

Oct 19, 2023

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, 2023

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.

Oct 19, 2023

Integral

A definite integral.

Oct 19, 2023

Derivative

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

A derivative conveyance

A derivative word

Oct 19, 2023

Integral

An indefinite integral.

Oct 19, 2023

Derivative

Imitative of the work of someone else.

Oct 19, 2023

Integral

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

Oct 19, 2023

Derivative

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

Oct 19, 2023

Integral

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

Oct 19, 2023

Derivative

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

Oct 19, 2023

Integral

(mathematics) Relating to integration.

Oct 19, 2023

Derivative

Lacking originality.

Oct 19, 2023

Integral

(obsolete) Whole; undamaged.

Oct 19, 2023

Derivative

Something derived.

Oct 19, 2023

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, 2023

Derivative

(linguistics) A word that derives from another one.

Oct 19, 2023

Integral

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

Oct 19, 2023

Derivative

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

Oct 19, 2023

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

\frac{1}{x}

on

[\frac{1}{2}, 1]

is

\ln(2)

, but the integral of the same function on

(0, 1]

diverges. In notation,

\int_\frac{1}{2}^1\frac{1}{x} dx = \ln(2)

, but

\int_0^1\frac{1}{x} dx = \infty

.

Oct 19, 2023

Derivative

(chemistry) A chemical derived from another.

Oct 19, 2023

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

\frac{x^3}{3}

plus a constant.

Oct 19, 2023

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, 2023

Integral

The fluent of a given fluxion in Newtonian calculus.

Oct 19, 2023

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'(x)

or

\frac{df}{dx}

in Leibniz's notation,

\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'(x) = 2x

Oct 19, 2023

Integral

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

A local motion keepeth bodies integral.

Oct 19, 2023

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.

Oct 19, 2023

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.

Oct 19, 2023

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, 2023

Integral

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

Oct 19, 2023

Derivative

(generally) The linear operator that maps functions to their derived functions, usually written

D

; the simplest differential operator.

Oct 19, 2023

Integral

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

Oct 19, 2023

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.

Oct 19, 2023

Integral

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

Oct 19, 2023

Derivative

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

Oct 19, 2023

Integral

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

Oct 19, 2023

Derivative

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

Oct 19, 2023

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

Oct 19, 2023

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.

Oct 19, 2023

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

Oct 19, 2023

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.

Oct 19, 2023

Integral

Essential or fundamental.

Trust is an integral part of a successful team.

Dec 26, 2023

Derivative

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

Oct 19, 2023

Derivative

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

Oct 19, 2023

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.

Oct 19, 2023

Derivative

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

Oct 19, 2023

Derivative

A financial instrument whose value is based on another security

Oct 19, 2023

Derivative

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

`electricity' is a derivative of `electric'

Oct 19, 2023

Derivative

Resulting from or employing derivation;

A derivative process

A highly derivative prose style

Oct 19, 2023

Derivative

A financial security whose value is dependent on an underlying asset.

The trader invested in a derivative based on commodity prices.

Dec 26, 2023

Derivative

Derived from something else; not original.

The artist's work was criticized as being derivative of earlier styles.

Dec 26, 2023

Derivative

Something that has been derived.

The study focused on the derivative forms of the language.

Dec 26, 2023

Repeatedly Asked Queries

Is a derivative always a slope?

In the context of a graph, it represents the slope at a point.

Dec 26, 2023

What does an integral represent?

It represents the total accumulation, like area under a curve.

Dec 26, 2023

What symbol is used for an integral?

The integral symbol ∫.

Dec 26, 2023

What is a derivative in calculus?

It's the rate of change of a function at a given point.

Dec 26, 2023

Can derivatives be applied in physics?

Yes, they're used to find rates like velocity and acceleration.

Dec 26, 2023

Can integrals determine displacement?

Yes, by integrating velocity over time.

Dec 26, 2023

Can integrals be used in probability?

Yes, especially in calculating cumulative distributions.

Dec 26, 2023

Is 'integral' used in non-math contexts?

Yes, to mean essential or fundamental.

Dec 26, 2023

Are integrals used in engineering?

Yes, for calculations like area and volume.

Dec 26, 2023

Are derivatives and integrals opposites?

They are inverse processes in calculus.

Dec 26, 2023

What does it mean when an integral is negative?

It indicates the accumulation below the x-axis.

Dec 26, 2023

Can integrals compute volumes?

Yes, by integrating cross-sectional areas.

Dec 26, 2023

Do integrals have geometric interpretations?

Yes, like area under a curve.

Dec 26, 2023

Do derivatives have real-world applications?

Yes, in fields like economics, physics, and engineering.

Dec 26, 2023

What's a derivative in finance?

A financial instrument based on another asset.

Dec 26, 2023

How are derivatives used in optimization?

To find maxima and minima of functions.

Dec 26, 2023

Are derivatives used in machine learning?

Yes, in algorithms like gradient descent.

Dec 26, 2023

Can a derivative be zero?

Yes, indicating no change at that point.

Dec 26, 2023

Can both concepts be applied in economics?

Yes, in analyzing rates of change and total quantities.

Dec 26, 2023

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Abstract vs. Conceptual

Derivative vs. Integral

Derivative vs. Integral

About Author

Written by

Shumaila Saeed, an expert content creator with 6 years of experience, specializes in distilling complex topics into easily digestible comparisons, shining a light on the nuances that both inform and educate readers with clarity and accuracy.

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