A partial derivative is a mathematical concept used in calculus to describe the rate of change of a function with respect to one of its variables while holding all other variables constant. It is denoted by ∂ and is used to calculate how a function changes in response to an infinitesimal change in one of its variables.
A regular derivative, denoted by d, calculates the rate of change of a function with respect to a single variable. A partial derivative, on the other hand, calculates the rate of change with respect to one variable while holding all other variables constant. This allows us to study the behavior of a function in multiple dimensions.
Partial derivatives are used in many fields, including physics, engineering, economics, and statistics. They are particularly useful in multivariable calculus, where they are used to optimize functions with multiple variables and to describe the behavior of complex systems.
To calculate a partial derivative, you first take the derivative of the function with respect to the variable of interest, treating all other variables as constants. This results in a new function with the variable of interest as the independent variable. Then, you plug in the specific values of the other variables to get the final result.
While a partial derivative calculates the rate of change with respect to one variable while holding all other variables constant, a total derivative calculates the overall change of a function with respect to all of its variables. In other words, a partial derivative gives a local picture of how a function changes, while a total derivative gives a global picture.