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...

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...

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...

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

symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate...

{\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...

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

{\frac {\partial L}{\partial f'}}(a)\delta f(a)\end{aligned}}} 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...

this extremum solution corresponds to the minimum from the second partial derivative test by noting that the variance is a quadratic function of the weights...

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

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

the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued...

consider the operators, denoted ∂ i {\displaystyle \partial _{i}} , that perform directional derivatives in the directions of e i {\displaystyle e_{i}} :...

{\partial (u_{1},\ldots ,u_{m})}{\partial (x_{1},\ldots ,x_{n})}}.} The chain rule for total derivatives implies a chain rule for partial derivatives....

the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described...

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

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...

which partial derivatives exist in every direction but fail to be differentiable. Furthermore, this definition as the vector of partial derivatives is only...

function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals...

Sonin–Letnikov derivative Liouville derivative Caputo derivative Hadamard derivative Marchaud derivative Riesz derivative Miller–Ross derivative Weyl derivative Erdélyi–Kober...

notation in a given context. For more specialized settings—such as partial derivatives in multivariable calculus, tensor analysis, or vector calculus—other...

{\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}f={\frac {\partial f}{\partial x}}\mathbf...

f , {\displaystyle T=\partial ^{p}f,} where the derivatives are understood in the sense of distributions. That is, for all test functions ϕ {\displaystyle...

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...

mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René...

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h ( x ) = 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...