When differentiating a function f of a variable x, one examines the change in behaviour of fin the direction of x. For a scalar field f in the n variables \(x_1 ,\dots , x_n\) , there are many directions in which the function can change. The partial derivatives provide this change in behaviour in the directions of the axes, the directional derivative much more generally in any arbitrary direction. Fortunately, this partial differentiation (and also the formation of the directional derivative) does not bring any new difficulties: one simply differentiates with respect to the considered variable, as one is used to from the one-dimensional case, and freezes all other variables. In this way, we easily obtain the gradient as a collection of the first partial derivatives, and the Hessian matrix as a collection of the second partial derivatives of a scalar field f and the Jacobian matrix as a collection of the first partial derivatives of a vector-valued function in several variables.