Derivative vs. Integral — What's the Difference?

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

Definite integrals have limits of integration, specifying the interval over which to integrate.

Higher-order derivatives represent rates of change of derivative functions.

An integral sums the infinite infinitesimal parts of a function to find whole areas or volumes.

Derivatives are crucial for finding local maxima and minima of functions.

Integrals are essential in computing total quantities, such as mass or charge over an object.

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.

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.

Resulting from or employing derivation

Necessary to make a whole complete; essential or fundamental

Copied or adapted from others

Of or denoted by an integer.

Something derived.

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.

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

Essential or necessary for completeness; constituent

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

Possessing everything essential; entire.

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

Expressed or expressible as or in terms of integers.

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

Expressed as or involving integrals.

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

A complete unit; a whole.

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.

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.

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

A definite integral.

Imitative of the work of someone else.

An indefinite integral.

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

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

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

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

Lacking originality.

(mathematics) Relating to integration.

Something derived.

(obsolete) Whole; undamaged.

(linguistics) A word that derives from another one.

(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.

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

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

(chemistry) A chemical derived from another.

(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.

(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.

(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 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 fluent of a given fluxion in Newtonian calculus.

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.

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

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

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

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

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

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

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

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

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

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

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

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.

Existing as an essential constituent or characteristic;

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.

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

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

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

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.

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

A financial instrument whose value is based on another security

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

Resulting from or employing derivation;

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

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

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

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

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

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

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

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

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

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

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

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

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