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

Edited by Tayyaba Rehman — By Maham Liaqat — Updated on April 17, 2024
Derivatives measure the rate of change of functions, crucial in understanding dynamics, while integrals compute the area under curves, essential for aggregation.

## Key Differences

Derivatives focus on the instantaneous rate of change at any given point of a function, essentially determining how a function changes as its input changes. In contrast, integrals are used to calculate the total accumulation of quantities, which can be thought of as the area under a curve between two points on a graph.
Derivatives are fundamental in fields like physics and engineering, where changes in motion or rates are crucial, such as calculating velocity or acceleration from a position-time graph. On the other hand, integrals are key in processes involving total quantities, such as finding the total distance traveled given a velocity-time graph.
The process of finding a derivative is known as differentiation, which involves applying specific rules to a function to determine another function that predicts changes. Whereas, integration, the process of finding an integral, often requires summing infinite slices to find whole values, which can be more computationally intensive.
Applications of derivatives include optimization problems, where one needs to find maximum or minimum values, crucial in economics and logistics. Conversely, integrals are used extensively in probability to determine the likelihood of events, integral to statistics and risk assessment.

## Comparison Chart

### Definition

Measures rate of change at a specific point.
Calculates total accumulation over an interval.

Differentiation
Integration

### Key Applications

Physics, engineering optimization
Probability, total distance or area calculations

### Complexity

Often straightforward with basic functions.
May require advanced techniques for complex functions.

## Compare with Definitions

#### Derivative

Derivatives can be used to find the slope of the tangent line at any point of a curve.
The slope of the tangent line to the curve at any point is the derivative at that point.

#### Integral

Definite integrals have limits of integration, specifying the interval over which to integrate.
The area under the curve from f(x)dx.

#### Derivative

Higher-order derivatives represent rates of change of derivative functions.
The second derivative of x is 2, indicating a constant rate of acceleration.

#### Integral

An integral sums the infinite infinitesimal parts of a function to find whole areas or volumes.
The integral of x from 0 to 1 is 0.5, representing the area under the line from x=1.

#### Derivative

Derivatives are crucial for finding local maxima and minima of functions.
The derivative set to zero can indicate potential maxima or minima in a function's graph.

#### Integral

Integrals are essential in computing total quantities, such as mass or charge over an object.
The total charge along a rod can be found by integrating the charge density over its length.

#### 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.

#### 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.

#### Derivative

Resulting from or employing derivation
A derivative word.
A derivative process.

#### 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

#### Derivative

A highly derivative prose style.

#### Integral

Of or denoted by an integer.

#### Derivative

Something derived.

#### 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.

#### Derivative

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

#### Integral

Essential or necessary for completeness; constituent
The kitchen is an integral part of a house.

#### Derivative

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

#### Integral

Possessing everything essential; entire.

#### Derivative

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

#### Integral

Expressed or expressible as or in terms of integers.

#### Derivative

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

#### Integral

Expressed as or involving integrals.

#### Derivative

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

#### Integral

A complete unit; a whole.

#### 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.

#### 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.

#### Derivative

Obtained by derivation; not radical, original, or fundamental.
A derivative conveyance
A derivative word

#### Integral

A definite integral.

#### Derivative

Imitative of the work of someone else.

#### Integral

An indefinite integral.

#### Derivative

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

#### Integral

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

#### Derivative

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

#### Integral

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

#### Derivative

Lacking originality.

#### Integral

(mathematics) Relating to integration.

#### Derivative

Something derived.

#### Integral

(obsolete) Whole; undamaged.

#### Derivative

(linguistics) A word that derives from another one.

#### 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!

#### Derivative

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

#### Integral

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

#### Derivative

(chemistry) A chemical derived from another.

#### 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 $\frac\left\{1\right\}\left\{x\right\}$ on $\left[\frac\left\{1\right\}\left\{2\right\}, 1\right]$ is $\ln\left(2\right)$, but the integral of the same function on $\left(0, 1\right]$ diverges. In notation, $\int_\frac\left\{1\right\}\left\{2\right\}^1\frac\left\{1\right\}\left\{x\right\} dx = \ln\left(2\right)$, but $\int_0^1\frac\left\{1\right\}\left\{x\right\} dx = \infty$.

#### 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.

#### 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 $\frac\left\{x^3\right\}\left\{3\right\}$ plus a constant.

#### Derivative

The derived function of $f\left(x\right)$: 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 $f\text{'}\left(x\right)$ or $\frac\left\{df\right\}\left\{dx\right\}$ in Leibniz's notation, $\dot\left\{f\right\}\left(x\right)$ 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 $f\left(x\right) = x^2$, then $f\text{'}\left(x\right) = 2x$

#### Integral

The fluent of a given fluxion in Newtonian calculus.

#### 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 $f\left(x\right)=x^3$ at $x=2$ is 12.

#### Integral

Lacking nothing of completeness; complete; perfect; uninjured; whole; entire.
A local motion keepeth bodies integral.

#### 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.

#### 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.

#### Derivative

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

#### Integral

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

#### 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.

#### Integral

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

#### Derivative

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

#### Integral

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

#### Derivative

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

#### Integral

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

#### 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.

#### 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

#### 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.

#### 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

#### Derivative

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

#### Derivative

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

#### 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.

#### Derivative

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

#### Derivative

A financial instrument whose value is based on another security

#### Derivative

(linguistics) a word that is derived from another word;
electricity' is a derivative of electric'

#### Derivative

Resulting from or employing derivation;
A derivative process
A highly derivative prose style

## Common Curiosities

#### How are integrals applied in real-world scenarios?

Integrals calculate total quantities like area, volume, and other accumulations, useful in engineering and physics.

#### How are derivatives used in real life?

Derivatives are used in physics, economics, and engineering to model changes and optimize outcomes.

#### What is differentiation?

Differentiation is the process of finding a derivative, determining how a function changes as its inputs change.

#### What is a derivative?

A derivative is a measure of how a function's output changes as the input changes.

#### What is the relationship between derivatives and integrals?

Derivatives and integrals are inverse processes; integrals can reverse the operation of differentiation.

#### Can you give an example of a derivative in a physical context?

The derivative of the position of a moving object with respect to time is its velocity.

#### What is an integral?

An integral represents the total accumulation of a function's values over an interval.

#### Can you provide an example of an integral used in physics?

Calculating the total work done by a force over a distance requires integrating the force over that distance.

#### How does one compute a derivative?

By applying rules of differentiation like the power rule, product rule, or chain rule to a function.

#### What are higher-order derivatives?

Higher-order derivatives are derivatives of derivatives, providing deeper insights into the changes of rates.

#### What is integration?

Integration is the process of calculating an integral, summing the parts of a function to find total values.

#### What are definite and indefinite integrals?

Definite integrals compute the total value over an interval with specific limits, whereas indefinite integrals do not have set limits.

#### How does one calculate an integral?

Through methods like substitution or parts, depending on the function's complexity and the desired precision.

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