What is the Difference Between Derivative and Differential?

The main difference between a derivative and a differential lies in their representation and purpose in calculus. Here are the key differences:

• Derivative: A derivative is a measure of how much a function changes with respect to its input. It represents the rate of change of the output value (y) with respect to its input variable (x). The derivative of a function is expressed as $$dy/dx$$ or $$f'(x)$$, and it is a function of only x. The process of finding a derivative is called differentiation.
• Differential: A differential is an infinitesimal change in a variable. It represents the actual change in the function. The differential of a function is given by $$df(x) = f'(x) dx$$, which is a function of both x and dx. Differentials are used in applications of calculus to approximate changes in a function and in optimization problems to find the maximum or minimum value of a function.

To summarize:

• A derivative is the rate at which a function changes at a particular point, represented as $$dy/dx$$ or $$f'(x)$$.
• A differential is the actual change in the function, represented as $$df(x) = f'(x) dx$$.

Both derivatives and differentials are essential concepts in calculus, but they serve different purposes and have distinct representations.

Comparative Table: Derivative vs Differential

The main difference between a derivative and a differential lies in their definitions and applications. Here is a table highlighting the key differences between the two:

In summary, a derivative is a measure of the rate of change of a function, while a differential represents the actual change in the function's output value with respect to its input. Both concepts are related but serve different purposes in mathematics and its applications.