What is a Total Derivative?

In multivariable calculus, the total derivative of a function is a way to generalize the concept of the derivative to functions of multiple variables. Total derivative of a function is the measure of the change in the dependent variable due to changes in all the independent variables.

It is an approximation of the actual change in the function and is used to determine the total change in the dependent variable when all the independent variables are varied.

Formula for Total Derivative

Suppose f is a function of n variables, x₁, x₂, . . . , x_n: f = f(x₁, x₂, . . . , x_n)

The total derivative of f with respect to a variable t (which could represent time or another parameter) is given by:

[Tex]\frac{df}{dt} = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i} \frac{dx_i}{dt}[/Tex]

Here:

• [Tex]\frac{\partial f}{\partial x_i}[/Tex]​ is the partial derivative of f with respect to x_i​.
• [Tex]\frac{dx_i}{dt}[/Tex] is the derivative of​ x_i with respect to t.

Total Derivative of a function measures how that function changes as all of its input variables change. For function f at a point is an approximation near the point of the function w.r.t. (with respect to) its arguments (variables). The total derivative never approximates the function with a single variable if two or more variables are present in the function.

Sometimes, the Total derivative is the same as the partial derivative or ordinary derivative of the function. In this article, we will discuss about total derivative in detail.