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 needs to be noted. Derivatives connote this rate of change by studying the slope of the function on a graph.
Differential vs Derivative
The main 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 quantities that are variable like the area of a body. It enables the calculation of the relationship between the independent and dependent variables in the equation.
Comparison Table Between Differential and Derivative
| 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 | Differential equations are represented by many formulas. 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 infinitesimal 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 are used to 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 representational equation. Differential equations are often represented as:
db/dy = f(a)
Where b is the dependent and a the independent variable.
What is a Derivative?
In the simplest of 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 time period, derivatives measure the steepness of the slope of a function.
Much like differentials, derivatives can also be classified as first-order and second-order derivatives. 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 with respect to 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). Derivatives are often used by scientific researchers in differential equations to gauge the changes in the value of variables to be able to succinctly predict the behavior of changing systems.
Main Differences Between Differentials and Derivatives
- The main difference between differentials and derivatives is in terms of their definitions which thereby impact their functionality in the mathematical realm. The former is a subdomain of calculus that connotes the infinitesimal difference in some fluctuating quantity. Derivatives, on the other hand, refer to the alteration of the output value due to a corresponding change in the input value. It connotes the rate of this change.
- Differentials equations contain derivatives or functions of derivations. Whereas, derivatives simply 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.
- 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.
- The formulas of a differential and derivative equation are also significantly different. dy/dx = f(x) represents the former, where y is the dependent and x the independent variable. Derivatives are represented by d/dx.
- 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.
Conclusion
Both differentials and derivatives are seminal mathematical concepts that are indispensable in the application and study of complex mathematical problems. They are both often used in conjunction with each other and can often be misinterpreted- if their meanings or functions remain unclear.
The differences between the two concepts are minimal but at the same time important to be cognized. The two concepts differ in terms of their implementation and usage in equations. While a differential equation contains derivatives or functions of derivatives, derivatives are the measure of instant change that occurs in a dependent variable that is triggered by a corresponding change in the independent variable.
Differentials are representational of the relationship that exists between two variables. They use derivatives to clearly define this relationship and measure infinitesimal changes.
The representation of each differs significantly. Moreover, differentials map real value alteration through linear mapping while derivatives map the slope of change. Each concept also embodies significant variable forms.
References
- https://ieeexplore.ieee.org/abstract/document/8579172/
- https://www.tandfonline.com/doi/abs/10.4169/074683410X480195