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What is a Total Derivative?

Total Derivative

A total derivative is a concept in multivariable calculus that generalizes the derivative of a single-variable function to a function of multiple variables. It captures the rate of change of the dependent variable with respect to an independent variable while considering the dependencies of the other variables.

In the context of a function with multiple variables, the total derivative provides a linear approximation of the function in a neighborhood of a given point. It is closely related to the partial derivative, which measures the rate of change of a function with respect to one variable while holding the others constant.

To compute the total derivative of a function f(x, y) with respect to an independent variable t, one can use the chain rule, a fundamental rule in calculus. The chain rule states that the total derivative df/dt is equal to the sum of the partial derivatives of f with respect to each variable, multiplied by the derivative of each variable with respect to t:

df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

The total derivative is particularly useful in the study of functions that describe physical processes, such as the motion of an object in a gravitational field, where the position and velocity of the object depend on multiple variables, like time and mass.

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