# Difference Between Differential and Derivative

To better understand the difference between the differential and derivative of a function, you need to understand the concept of a function first.

A function is one of the basic concepts in mathematics that defines a relationship between a set of inputs and a set of possible outputs where each input is related to one output. One variable is the independent variable and the other variable is the dependent variable.

The concept of function is one of the most underrated topics in mathematics but is essential in defining physical relationships. Take for example: the statement “y is a function of x” means something related to y is directly related to x by some formula. Let’s say if the input is 6 and the function is to add 5 to input 6. The result will be 6+5 = 11, which is your output.

There are few exceptions in mathematics or you can say problems, which cannot be solved by ordinary methods of geometry and algebra alone. A new branch of mathematics known as calculus is used to solve these problems.

Calculus is fundamentally different from mathematics which not only uses the ideas from geometry, arithmetic, and algebra, but also deals with change and motion.

The calculus as a tool defines the derivative of a function as the limit of a particular kind. The concept of derivative of a function distinguishes calculus from other branches of mathematics. Differential is a subfield of calculus that refers to infinitesimal difference in some varying quantity and is one of the two fundamental divisions of calculus. The other branch is called integral calculus.

## What is Differential?

Differential is one of the fundamentals divisions of calculus, along with integral calculus. It is a subfield of calculus that deals with infinitesimal change in some varying quantity. The world we live in is full of interrelated quantities that change periodically.

For example, the area of a circular body which changes as the radius changes or a projectile which changes with the velocity. These changing entities, in mathematical terms, are called as variables and the rate of change of one variable with respect to another is a derivative. And the equation which represents the relationship between these variables is called a differential equation.

Differential equations are equations that contain unknown functions and some of their derivatives.

## What is Derivative?

The concept of derivative of a function is one of the most powerful concepts in mathematics. The derivative of a function is usually a new function which is called as the derivative function or the rate function.

The derivative of a function represents an instantaneous rate of change in the value of a dependent variable with respect to the change in value of the independent variable. It’s a fundamental tool of calculus which can also be interpreted as the slope of the tangent line. It measures how steep the graph of a function is at some given point on the graph.

In simple terms, derivative is the rate at which function changes at some particular point.

## Difference between Differential and Derivative

### Definition of Differential Vs. Derivative

Both the terms differential and derivative are intimately connected to each other in terms of interrelationship. In mathematics changing entities are called variables and the rate of change of one variable with respect to another is called as a derivative.

Equations which define relationship between these variables and their derivatives are called differential equations. Differentiation is the process of finding a derivative. The derivative of a function is the rate of change of the output value with respect to its input value, whereas differential is the actual change of function.

### Relationship of Differential Vs. Derivative

Differentiation is a method of computing a derivative which is the rate of change of the output y of the function with respect to the change of the variable x.

In simple terms, derivative refers to the rate of change of y with respect of x, and this relationship is expressed as y = f(x), which means y is a function of x. Derivative of the function f(x) is defined as the function whose value generates the slope of f(x) where it is defined and f(x) is differentiable. It refers to the slope of the graph at a given point.

### Representation of Differential Vs. Derivative

Differentials are represented as *d*x, *d*y, *d*t, and so on, where *d*x represents a small change in x, *d*y represents a small change in y, and *d*t is a small change in t. When comparing changes in related quantities where y is the function of x, the differential *d*y can be written as:

*d*y = f^{’}(x) *d*x

The derivative of a function is the slope of the function at any point and is written as *d*/*d*x. For example, the derivative of sin(x) can be written as:

*d*/*d*x sin(x) = sin(x)^{’} = cos(x)

### Differential vs. Derivative: Comparison Chart

## Summary of Differential Vs. Derivative

In mathematics, the rate of change of one variable with respect to another variable is called a derivative and the equations which express relationship between these variables and their derivatives are called differential equations. In a nutshell, differentia equations involve derivatives which in fact specify how a quantity changes with respect to another. By solving a differential equation, you get a formula for the quantity that doesn’t contain derivatives. The method of computing a derivative is called differentiation. In simple terms, the derivative of a function is the rate of change of the output value with respect to its input value, whereas differential is the actual change of function.

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## References :

[0]Rohde, Ulrich, et al. Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners. Hoboken, New Jersey: John Wiley & Sons, 2012. Print

[1]Tenenbaum, Morris and Harry Pollard. Ordinary Differential Equations. Massachusetts: Courier Corporation, 1985. Print

[2]Courant, Richard. Differential and Integral Calculus. Hoboken, New Jersey: John Wiley & Sons, 2012. Print

[3]Image credit: https://en.wikipedia.org/wiki/Differential_item_functioning#/media/File:Nonuni_DIF_ICC.png

[4]Image credit: https://upload.wikimedia.org/wikipedia/commons/f/f1/Detachment_derivative_dy.png