Derivative (multivariable calculus)

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In mathematics, the derivative of a function at a point is the linear part of the best affine approximation to the function near the point. In one-variable calculus, this is the tangent line approximation. In multivariable calculus, the same property is generalized to define the derivative of a vector-valued function or function of a vector argument. Sometimes called the total derivative, in contrast with partial derivatives, the 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. In functional analysis, particularly in infinite dimensions, the derivative in this sense is called the Fréchet derivative.

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