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Concept
Total derivative
Summary
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the ordinary derivative of the function.
Let be an open subset. Then a function is said to be (totally) differentiable at a point if there exists a linear transformation such that
The linear map is called the (total) derivative or (total) differential of at . Other notations for the total derivative include and . A function is (totally) differentiable if its total derivative exists at every point in its domain.
Conceptually, the definition of the total derivative expresses the idea that is the best linear approximation to at the point . This can be made precise by quantifying the error in the linear approximation determined by . To do so, write
where equals the error in the approximation. To say that the derivative of at is is equivalent to the statement
where is little-o notation and indicates that is much smaller than as . The total derivative is the unique linear transformation for which the error term is this small, and this is the sense in which it is the best linear approximation to .
The function is differentiable if and only if each of its components is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain. However, the same is not true of the coordinates in the domain. It is true that if is differentiable at , then each partial derivative exists at . The converse does not hold: it can happen that all of the partial derivatives of at exist, but is not differentiable at .
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