Difference Between Differential and Derivative

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,

Now let's learn about, Differential and Derivative in brief

What is Differential?

In mathematics, the term "differential" refers to an infinitesimal change in a function or variable. It represents a small or incremental change in a quantity or function. 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 respect to 𝑥 is represented as "dy" or "δy" (delta y). Similarly, the differential of 𝑥 is denoted as "dx" or "δx" (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 functions at specific points and study their local changes.

Example:

Suppose 𝑦 =𝑓(𝑥) The differential of 𝑦 denoted as 𝑑𝑦 is given by:

𝑑𝑦 =𝑓′(𝑥)

Here, 𝑑𝑥 represents an infinitesimal change in 𝑥 and 𝑓′(𝑥) is the derivative of 𝑓 with respect to 𝑥 The differential 𝑑𝑦 approximates the change in 𝑦 for a small change in 𝑥.

What is Derivative?

In mathematics, the derivative is a fundamental concept in 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 (Δy) to the change in the independent variable (Δx) as the interval between the two points approaches zero:

f ′(x) = lim Δx→0 (x+Δx)−f(x) / Δx

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.

Read More,

• Derivatives
• Differential Equations
• Differentiation and Integration Formula