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In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. [citation needed]The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction ...
Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus , Newton's method (also called Newton–Raphson ) is an iterative method for finding the roots of a differentiable function f {\displaystyle f} , which are solutions to the equation f ( x ) = 0 {\displaystyle f(x)=0} .
Steps one and two need only be carried out once, and tend to be costly. When one wants to calculate the Hessian at numerous points (such as in an optimization routine), steps 3 and 4 are repeated. Coloured Sparsity pattern of the Hessian Matrix
The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum.
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}
If all the partial derivatives of a function exist in a neighborhood of a point x 0 and are continuous at the point x 0, then the function is differentiable at that point x 0. However, the existence of the partial derivatives (or even of all the directional derivatives ) does not guarantee that a function is differentiable at a point.
Thus we have a formula for ∂ v f, (one of ways to represent the directional derivative) where v is arbitrary; for ():= [,] (see its full definition above), its directional derivative with respect to v is = = = where the first two equalities just show different representations of the directional derivative.
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.