Second partial derivative test information

In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local...

the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can...

In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local...

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held...

)}{\partial \mathbf {x} }}.} It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear...

of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of...

f(b)\,-\,{\frac {\partial L}{\partial f'}}(a)\delta f(a)\,} where the variation in the derivative, δf ′ was rewritten as the derivative of the variation...

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...

Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes...

minimum, or neither by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability. For...

{\displaystyle {\frac {\partial f}{\partial x}}=2x+y,\qquad {\frac {\partial f}{\partial y}}=x+2y.} In general, the partial derivative of a function f ( x...

}={\frac {\partial g}{\partial x^{i}}}\,dx^{i}\wedge dx^{I}} (using the Einstein summation convention). The definition of the exterior derivative is extended...

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable...

function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the...

calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a...

{\partial ^{2}f}{\partial x\,\partial y}},\\[5pt]&\partial _{yy}f={\frac {\partial ^{2}f}{\partial y^{2}}}.\end{aligned}}} See § Partial derivatives. D-notation...

the derivative, one repeatedly applies partial derivatives with respect to different variables. For example, the second order partial derivatives of a...

differentiation Stationary point Maxima and minima First derivative test Second derivative test Extreme value theorem Differential equation Differential...

The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives...

formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely...

monotonicity is not present and we cannot apply the test. Actually the series is divergent. Indeed, for the partial sum S 2 n {\displaystyle S_{2n}} we have S...

{\displaystyle {\frac {\partial ^{\alpha }u}{\partial t^{\alpha }}}=-K(-\Delta )^{\beta }u.} A simple extension of the fractional derivative is the variable-order...

position, the gradient is given by the vector whose components are the partial derivatives of f {\displaystyle f} at p {\displaystyle p} . That is, for f :...

_{a(x)}^{b(x)}{\frac {\partial }{\partial x}}f(x,t)\,dt\end{aligned}}} where the partial derivative ∂ ∂ x {\displaystyle {\tfrac {\partial }{\partial x}}} indicates...

cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in...

series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the...

Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated...