Differential vs. Derivative — What's the Difference?
By Tayyaba Rehman — Updated on October 13, 2023
A differential represents a change in a function, while a derivative measures the rate of that change. Both are foundational concepts in calculus.
Difference Between Differential and Derivative
Table of Contents
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Key Differences
Differential and Derivative: A differential, symbolized as "dx" or "dy," indicates a small change in a variable. In contrast, a derivative indicates how one variable changes in relation to another, often symbolized as "dy/dx."
Differential and Derivative: While the differential provides a measure of the change in the function's value, the derivative gives the slope of the tangent line to a function at a point, representing the function's rate of change.
Differential and Derivative: Differentials are often used in approximations, especially in linear approximations, to indicate small changes. Derivatives, on the other hand, form the basis for understanding how functions behave and vary.
Differential and Derivative: In applications, a differential can describe incremental changes, like the difference in height or distance. A derivative would express how these quantities change concerning another, like time.
Differential and Derivative: From an operational viewpoint, finding the differential involves understanding the changes between two points. Finding the derivative requires understanding the instantaneous rate of change at a singular point.
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Comparison Chart
Nature
Represents a small change in a function.
Measures the rate of change of a function.
Symbolization
Often "dx" or "dy".
Typically "dy/dx" or "f'(x)".
Application
Used for approximations of changes.
Describes the behavior and variation of functions.
Representation
Describes incremental changes.
Expresses how quantities change relative to others.
Operational View
Understanding changes between points.
Understanding instantaneous rate of change at a point.
Compare with Definitions
Differential
The difference or disparity between things.
The differential in skills between the two players was evident.
Derivative
Something derived or obtained from a source.
The musical piece was a derivative of an older composition.
Differential
A small change in a function or variable.
The differential in temperature was only a few degrees.
Derivative
Lacking originality or being imitative.
Critics called the movie derivative and uninspired.
Differential
A gear or system that allows for difference in wheel speed.
The car's differential helps it navigate turns effectively.
Derivative
The slope of the tangent line to a curve at a given point.
By studying the derivative, we can understand the function's behavior.
Differential
Of, relating to, or showing a difference.
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.
Differential
Constituting or making a difference; distinctive.
Derivative
Resulting from or employing derivation
A derivative word.
A derivative process.
Differential
Dependent on or making use of a specific difference or distinction.
Derivative
Copied or adapted from others
A highly derivative prose style.
Differential
(Mathematics) Of or relating to differentiation.
Derivative
Something derived.
Differential
Involving differences in speed or direction of motion.
Derivative
(Linguistics) A word formed from another by derivation, such as electricity from electric.
Differential
An infinitesimal increment in a variable.
Derivative
The limiting value of the ratio of the change in a function to the corresponding change in its independent variable.
Differential
The product of the derivative of a function of one variable and the increment of the independent variable.
Derivative
The instantaneous rate of change of a function with respect to its variable.
Differential
A differential gear.
Derivative
The slope of the tangent line to the graph of a function at a given point. Also called differential coefficient, fluxion.
Differential
A difference between comparable things, as in wage rate or in price.
Derivative
(Chemistry) A compound derived or obtained from another and containing essential elements of the parent substance.
Differential
Of or pertaining to a difference.
Differential characteristics
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.
Differential
Dependent on, or making a difference; distinctive.
Derivative
Obtained by derivation; not radical, original, or fundamental.
A derivative conveyance
A derivative word
Differential
Having differences in speed or direction of motion.
Derivative
Imitative of the work of someone else.
Differential
(mathematics) Of or pertaining to differentiation or the differential calculus.
Derivative
Referring to a work, such as a translation or adaptation, based on another work that may be subject to copyright restrictions.
Differential
The differential gear in an automobile, etc.
Derivative
(finance) Having a value that depends on an underlying asset of variable value.
Differential
A qualitative or quantitative difference between similar or comparable things.
Derivative
Lacking originality.
Differential
(mathematics) An infinitesimal change in a variable, or the result of differentiation.
Derivative
Something derived.
Differential
One of two coils of conducting wire so related to one another or to a magnet or armature common to both, that one coil produces polar action contrary to that of the other.
Derivative
(linguistics) A word that derives from another one.
Differential
A form of conductor used for dividing and distributing the current to a series of electric lamps so as to maintain equal action in all.
Derivative
(finance) A financial instrument whose value depends on the valuation of an underlying asset; such as a warrant, an option etc.
Differential
Relating to or indicating a difference; creating a difference; discriminating; special; as, differential characteristics; differential duties; a differential rate.
For whom he produced differential favors.
Derivative
(chemistry) A chemical derived from another.
Differential
Of or pertaining to a differential, or to differentials.
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.
Differential
Relating to differences of motion or leverage; producing effects by such differences; said of mechanism.
Derivative
The derived function of : the function giving the instantaneous rate of change of ; equivalently, the function giving the slope of the line tangent to the graph of . Written or in Leibniz's notation, in Newton's notation (the latter used particularly when the independent variable is time). Category:en:Functions
The derivative of is ; if , then
Differential
An increment, usually an indefinitely small one, which is given to a variable quantity.
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 at is 12.
Differential
A small difference in rates which competing railroad lines, in establishing a common tariff, allow one of their number to make, in order to get a fair share of the business. The lower rate is called a differential rate. Differentials are also sometimes granted to cities.
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.
Differential
One of two coils of conducting wire so related to one another or to a magnet or armature common to both, that one coil produces polar action contrary to that of the other.
Derivative
(generally) The linear operator that maps functions to their derived functions, usually written ; the simplest differential operator.
Differential
The result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx
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.
Differential
A quality that differentiates between similar things
Derivative
Hence, unoriginal (said of art or other intellectual products.
Differential
A bevel gear that permits rotation of two shafts at different speeds; used on the rear axle of automobiles to allow wheels to rotate at different speeds on curves
Derivative
That which is derived; anything obtained or deduced from another.
Differential
Relating to or showing a difference;
Differential treatment
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.
Differential
Involving or containing one or more derivatives;
Differential equation
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.
Differential
The difference between two values or quantities.
The price differential made one product more attractive than the other.
Derivative
An agent which is adapted to produce a derivation (in the medical sense).
Differential
In calculus, an infinitesimal change in a quantity.
When studying motion, we consider the differential of position.
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
Derivative
A measure of how a function changes as its input changes.
The derivative of velocity concerning time is acceleration.
Derivative
In finance, a contract whose value depends on underlying assets.
He invested in derivatives to diversify his portfolio.
Common Curiosities
Why do we study differentials in calculus?
Differentials are essential for understanding and approximating changes in functions.
How is a derivative defined?
A derivative measures the rate at which a function changes as its input varies.
Can a derivative be zero?
Yes, a derivative can be zero, often indicating a potential maximum or minimum of a function.
What does the differential "dx" mean?
"dx" typically represents an infinitesimally small change in the variable x.
Can a differential be negative?
Yes, a differential can be negative, indicating a decrease in the quantity.
How are differentials used in engineering?
Engineers use differentials to approximate changes and understand systems' behavior in various conditions.
What is a differential?
A differential represents a small change in a function or variable.
Can a differential be thought of as a difference?
Yes, a differential often represents the difference between two values or quantities.
What's the physical meaning of a derivative?
In physics, a derivative can represent rates of change, like velocity being the derivative of position concerning time.
What does "second derivative" mean?
The second derivative is the derivative of the first derivative and can indicate the curvature or concavity of a function.
Why is the concept of a derivative important?
Derivatives help understand how functions behave, optimize processes, and solve various practical problems.
Is every derivative a rate of change?
Yes, a derivative typically represents the rate of change of a function concerning its input.
How does one compute a derivative?
Derivatives are often computed using various rules and techniques of calculus, like the power rule or chain rule.
How do derivatives apply in economics?
In economics, derivatives can represent rates of change, like the change in cost concerning production units.
In what fields are derivatives crucial?
Derivatives are essential in physics, economics, biology, engineering, and various other fields.
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Written by
Tayyaba RehmanTayyaba Rehman is a distinguished writer, currently serving as a primary contributor to askdifference.com. As a researcher in semantics and etymology, Tayyaba's passion for the complexity of languages and their distinctions has found a perfect home on the platform. Tayyaba delves into the intricacies of language, distinguishing between commonly confused words and phrases, thereby providing clarity for readers worldwide.
