Differential vs Derivative: Difference and Comparison

Derivatives are contained within differential equations. They represent the rate of change of the variables. When the independent variable changes, the corresponding change produced in the dependent variable must be noted.

Derivatives connote this rate of change by studying the slope of the function on a graph.  

Key Takeaways

1. The derivative is a mathematical concept that describes the instantaneous rate of change of a function; the differential is a mathematical operator used to express the rate of change of a variable in relation to another variable.
2. The derivative is represented as the limit of the ratio of the change in the function to the change in the independent variable as the change in the independent variable approaches zero; the differential is expressed as the product of the derivative and the change in the independent variable.
3. The derivative is used to determine slopes and rates of change in calculus; the differential is used to solve differential equations and express the relationship between variables in physics and engineering.

Differential vs. Derivative

The difference between a differential and a derivative is in terms of the function each performs and the values each represents. Differentials represent the smallest of differences in variable quantities, like the area of a body. It enables calculating the relationship between the independent and dependent variables in the equation.

Comparison Table

Parameters of Comparison	Differentials	Derivatives
Definition	Differentials represent the smallest of differences in quantities that are variable.	Derivatives represent the rate of change of the variables in a differential equation.
Difference Calculated	The linear difference is calculated.	The slope of the graph at a particular point is calculated.
Relationship	Differential equations use derivatives to come to definitive solutions. Derivatives are contained within differential equations.	Derivatives simply connote the rate of change of the dependent variable vis-à-vis the independent variable.
Functional Connotations	Functional connotations between variables are unknown	Functional connotations between variables are known.
Represented By	Many formulas represent differential equations. One of the commonly used ones is: dy/dx = f(x)	There are various degrees of derivatives with diverse formulas of representation. The most commonly used formulaic representation of a derivative is: d/dx.

What is a Differential?

As a subfield of calculus, differential equations represent the tiny difference in certain fluctuating quantities. Differential equations contain derivatives and their functions.

Differentials measure the linear trajectory of change in the dependent variable as a consequence of altering the quantity of the independent variable. There are several different kinds of differential equations with varying orders and degrees of mathematical complexity.

Differential equations describe the movement of heat waves, the change in population numbers, the decay of radioactive material, electricity movement, the motion of a pendulum, etc.

Essentially differential equations connote the relationship between two variables, where the alteration of one variable is triggered by the change produced in the other.

It is the methodological tool used to calculate the derivatives of functions. Hence, it is a symbolic equation. Differential equations are represented as:

db/dy = f(a)

Where b is the dependent and the independent variable.

What is a Derivative?

In simplest terms, derivatives refer to the rate of change in variables when a change is recorded in the independent variable and a corresponding change is produced in the dependent variable. Hence, it highlights the change in output due to a change in the input value.

Derivatives are most commonly used with differential equations. Differentiation is the process used to find derivatives. They are used to connote the slope of a tangent line. Within a given period, derivatives measure the steepness of the slope of a function.

Like differentials, derivatives can also be classified as first- and second-order. While the former can be directly predicted from the slope of the line, the latter takes the concavity of the graph into account.

They are an important part of mathematical calculations. Often the slope is represented as:

d/dx

For instance, a derivation is defined as the rate of change of b concerning a. This relationship is expressed as b= f(a), where b is a function of a. The value of this function creates the slope of f(a).

Scientific researchers use derivatives in differential equations to gauge the changes in the value of variables to predict the behavior of changing systems succinctly.

Main Differences Between Differentials and Derivatives

1. The main difference between differentials and derivatives is their definitions, which impact their functionality in the mathematical realm. The former is a subdomain of calculus that connotes the infinitesimal difference in some fluctuating quantity. Derivatives, however, refer to altering the output value due to a corresponding change in the input value. It connotes the rate of this change.
2. Differential equations contain derivatives or functions of derivations. At the same time, derivatives refer to the instant change that occurs with the alteration of the independent variable that produces a corresponding change in the value of the dependent variable.
3. The functional connotation between the dependent and independent variables is known in the case of a derivative and unknown in the case of a differential. This represents another important difference between the two mathematical concepts.
4. The formulas of differential and derivative equations are also significantly different. dy/dx = f(x) represents the former, where y is the dependent and x is the independent variable. Derivatives are represented by d/dx.
5. Differentials represent the real value change through a linear map, while derivatives represent the same change through a slope map. Derivatives calculate the slope of a function on the graph at any given point in time.

References

1. https://ieeexplore.ieee.org/abstract/document/8579172/
2. https://www.tandfonline.com/doi/abs/10.4169/074683410X480195