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Difference Between Differential and Derivative
Differential and derivative are two fundamental concepts in calculus that are often used interchangeably. While they are related, they are not the same thing. Understanding the difference between differential and derivative is important in mastering calculus and its applications. In this guide, we will explore the difference between differential and derivative and their uses in calculus.
What are Differentials?
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 are Derivatives?
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, a derivative is the rate at which function changes at some particular point.
Differences: Differential and Derivative
While differential and derivative are related, they are not the same thing. 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 with respect to its input.
Another difference is that the differential is a function of two variables, while the derivative is a function of one variable. The differential of a function is given by df(x) = f'(x) dx, which is a function of both x and dx. The derivative, on the other hand, is given by f'(x) or dy/dx, which is a function of only x.
The differential is often used in applications of calculus to approximate changes in a function, while the derivative is used to find the rate of change of a function at a given point. The differential is also used in optimization problems to find the maximum or minimum value of a function, while the derivative is used in a variety of applications, including physics, economics, and engineering.
The following table highlights the major differences between Differentials and Derivatives:
Characteristics |
Differential |
Derivative |
|---|---|---|
Definition |
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. |
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. |
Relationship |
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 |
Differentials are represented as dx, dy, dt, and so on, where dx represents a small change in "x", dy represents a small change in "y", and dt is a small change in "t". When comparing changes in related quantities where y is the function of "x", the differential dy can be written as: dy = f’(x) dx |
The derivative of a function is the slope of the function at any point and is written as d/dx. For example, the derivative of sin(x) can be written as: d/dx sin(x) = sin(x)’ = cos(x) |
Conclusion
In conclusion, differential and derivative are two fundamental concepts in calculus that are often used interchangeably but are not the same thing. A differential is an infinitesimal change in a variable, while a derivative is a measure of how much the function changes with respect to its input. While both concepts are used in a variety of applications, it is important to understand the difference between them in order to apply them correctly.
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