# Difference Between Differential and Derivative

Last Updated : 23 Jan, 2024

Differential and Derivative both are related but they are not the same. The main difference between differential and derivative is that a differential is an infinitesimal change in a variable, while a derivative is a measure of how much the function changes for its input.

In this article, we will explore the difference between differential and derivative and understand their roles in calculus.

## Difference Between Differential and Derivative

The basic differences between Differential and Derivative are added in the table below,

Differential Vs Derivative

Differential

Derivative

Represents the infinitesimal change in a function or variable.

Represents the rate of change of a function at a point or over an interval.

Denoted by “d” followed by the variable.

Denoted by prime notation.

Measures the change in the dependent variable.

Measures the rate of change of the dependent variable.

Indicates the change in the function with respect to an independent variable.

Indicates the slope or gradient of the function at a specific point.

Can be used to calculate the total change by the integrating over an interval.

Can be used to find the slope of the tangent line or the instantaneous rate of change.

Represents the incremental or local change.

Represents the average or instantaneous change.

It is useful in physics, engineering and calculus.

Fundamental concepts in the calculus and mathematical analysis.

It enables calculation of the small changes in quantities.

It enables studying functions and their behavior

Now let’s learn about, Differential and Derivative in breif

## What is Differential?

In mathematics, the term “differential” refers to an infinitesimal change in the function or variable. It represents the small or incremental change in a quantity or function. The concept of differentials is closely related to calculus specifically differential calculus.

• Differentials are denoted by the symbol “d” followed by the variable. For example, if y is a function of x the differential of y with the respect to x is represented as “dy” or “Î´y” (delta y). Similarly, the differential of x is denoted as “dx” or “Î´x” (delta x).
• Differentials are used to approximate changes in functions especially when dealing with small increments.
• They play a significant role in calculus allowing for the calculation of rates of change finding tangent lines and solving optimization problems. By considering differentials. it becomes possible to analyze the behavior of the functions at specific points and study their local changes.

## What is Derivative?

In mathematics, the derivative is a fundamental concept in the calculus that represents the rate of change of a function with respect to its independent variable. It measures how a function’s output value changes when the input variable changes. The derivative provides information about the slope or gradient of a function at a particular point or over an interval.

The derivative of a function f(x) denoted as f'(x) or df/dx is defined as the limit of the ratio of the change in the function’s output (Î”y) to the change in independent variable (Î”x) as the interval between the two points approaches zero:

f'(x) = lim(Î”xâ†’ 0) [f(x + Î”x) – f(x)] / Î”x

Geometrically, the derivative represents the slope of the tangent line to the graph of the function at a given point. It provides information about function’s instantaneous rate of change at that point. The image for the same is added below,

The derivative has several interpretations and applications, including:

• Slope of a Tangent Line: The derivative gives the slope of the tangent line to graph of a function at a specific point.
• Rate of Change: The derivative represents the rate at which the dependent variable changes with the respect to independent variable.
• Optimization: The derivative helps in finding the maximum and minimum points of the function which is crucial in optimization problems.
• Velocity and Acceleration: In physics, the derivative is used to determine velocity and acceleration in kinematic problems.
• Shape of Curves: The derivative provides information about the concavity and convexity of a curve.

## Conclusion

In conclusion, differential and derivative are related concepts in calculus but serve different purposes. The differential represents an infinitesimal change in a function or variable and is denoted by “d” followed by the variable. It captures the small or incremental change in a quantity and is useful for the approximating changes in the functions and calculating total changes through integration. On the other hand, the derivative represents the rate of change of a function with the respect to its independent variable. It measures how the output value of the function changes as the input variable changes.

## Differential and Derivative – FAQs

### What is the difference between Derivative and Differential?

The main difference lies in their interpretation and purpose. A differential represents an infinitesimal change in the function or variable. whereas a derivative represents the rate of the change of a function at a specific point or over an interval.

### How are Differential and Derivative Represented Notationally?

Differentials are denoted by the symbol “d” followed by the variable while derivatives are typically represented using the prime notation.

### What do Differential and Derivative Measure?

Differentials measure the change in the dependent variable while derivatives measure the rate of the change of the dependent variable with respect to the independent variable.

### Is Differentiate the Same as Derivative?

Derivative of a function is the rate of change of a variable with respect the change of some other variable where as differentiation is the process of finding the derivative.

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