In mathematics, the gradient is a multi-variable generalization of the derivative. While a derivative can be defined on functions of a single variable, for functions of several variables, the gradient takes its place. The gradient is a vector-valued function, as opposed to a derivative, which is scalar-valued. If f(x1, ..., xn) is a differentiable, real-valued function of several variables, its gradient is the vector whose components are the n partial derivatives of f.