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Mathematical operation
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
Second_derivative
Mathematical theorem
symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate
Symmetry of second derivatives
Symmetry_of_second_derivatives
Instantaneous rate of change (mathematics)
and the second derivative is its acceleration. Derivatives can be generalized to functions of several real variables. In this case, the derivative is reinterpreted
Derivative
Method in multivariable calculus
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
Second partial derivative test
Second_partial_derivative_test
Derivative of a function with multiple variables
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_derivative
Method for finding the extrema of a function
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
Derivative_test
Matrix of second 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
Hessian_matrix
Notation of differential calculus
standard notation for differentiation. Instead, several notations for the derivative of a function or a dependent variable have been proposed by various mathematicians
Notation_for_differentiation
Mathematical identities
The following are important identities involving derivatives and integrals in vector calculus. For a function f ( x , y , z ) {\displaystyle f(x,y,z)}
Vector_calculus_identities
Rate of change of the second derivative
a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate
Third_derivative
Mathematical functions and constants
Explicit formulas for eigenvalues and eigenvectors of the second derivative with different boundary conditions are provided both for the continuous and
Eigenvalues and eigenvectors of the second derivative
Eigenvalues_and_eigenvectors_of_the_second_derivative
Higher derivatives of the position vector with respect to time
– with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively. These higher-order derivatives are less common than
Fourth, fifth, and sixth derivatives of position
Fourth,_fifth,_and_sixth_derivatives_of_position
Type of derivative in mathematics
In mathematics, the derivative of a function at a point is the linear part of the best affine approximation to the function near the point. In one-variable
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Instantaneous rate of change of the function
In multivariable calculus, the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given
Directional_derivative
Study of rates of change
differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen
Differential_calculus
Branch of mathematical analysis
Sonin–Letnikov derivative Liouville derivative Caputo derivative Hadamard derivative Marchaud derivative Riesz derivative Miller–Ross derivative Weyl derivative Erdélyi–Kober
Fractional_calculus
Derivative of a function with respect to time
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable
Time_derivative
Vector differential operator
function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function
Del
Piecewise function that clamps its input to be non-negative
is a Green's function for the second derivative operator. Thus, any function, f(x), with an integrable second derivative, f″(x), will satisfy the equation:
Ramp_function
Generalization of the concept of directional derivative
mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René
Gateaux_derivative
Point where the curvature of a curve changes sign
a function f of differentiability class C2 (its first derivative f', and its second derivative f'', exist and are continuous), the condition f'' = 0 can
Inflection_point
Formula in calculus
formula that expresses the derivative of the composition of two differentiable functions z and y in terms of the derivatives of z and y. More precisely
Chain_rule
Relation between vapour pressure and temperature
does not provide any information about its curvature or second derivative. The second derivative of the coexistence curve of phases 1 and 2 is given by
Clausius–Clapeyron_relation
Matrix of partial derivatives of a vector-valued function
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
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Mathematical operation in calculus
the logarithmic derivative of a function f is defined by the formula f ′ f {\displaystyle {\frac {f'}{f}}} where f′ is the derivative of f. Intuitively
Logarithmic_derivative
Operation on differential forms
the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described
Exterior_derivative
Laws in physics about force and motion
{v(t+\Delta t)-v(t)}{\Delta t}}.} Consequently, the acceleration is the second derivative of position, often written d 2 s d t 2 {\displaystyle {\frac {\mathrm
Newton's_laws_of_motion
Real function with secant line between points above the graph itself
twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex
Convex_function
Indefinite integral
inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal
Antiderivative
Differentiation under the integral sign formula
That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral
Leibniz_integral_rule
Operation in differential calculus
In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as: lim h → 0 f ( x + h ) − f ( x − h ) 2
Symmetric_derivative
Mathematical measure of how much a curve or surface deviates from flatness
{T}}(s)={\boldsymbol {\bar {\gamma }}}'(s).} If –γ is twice differentiable, the second derivative of –γ is T′(s), which is also the curvature vector, K(s). K ( s )
Curvature
Derivative defined on normed spaces
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
Fréchet_derivative
Branch of mathematics
derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative
Calculus
Negative of a convex function
{\displaystyle g(x)={\sqrt {x}}} are concave on their domains, as their second derivatives f ″ ( x ) = − 2 {\displaystyle f''(x)=-2} and g ″ ( x ) = − 1 4 x
Concave_function
Method for finding stationary points of a function
setting x k + 1 = x k + t {\displaystyle x_{k+1}=x_{k}+t} . If the second derivative is positive, the quadratic approximation is a convex function of t
Newton's method in optimization
Newton's_method_in_optimization
Largest and smallest value taken by a function at a given point
local minimum, or neither by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability
Maximum_and_minimum
Rules for computing derivatives of functions
a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are
Differentiation_rules
Mathematical operation in calculus
In calculus, implicit differentiation is a method for finding the derivative of a function that is defined by an equation rather than by an explicit formula
Implicit_differentiation
Type of financial contract
a derivative is a contract between a buyer and a seller. The derivative can take various forms, depending on the transaction, but every derivative has
Derivative_(finance)
Derivative in differential geometry and vector calculus
calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect
Second_covariant_derivative
Discrete analog of a derivative
difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f : Second-order central f ″ (
Finite_difference
Concept in differential calculus
In the calculus of variations, the second variation extends the idea of the second derivative test to functionals. Much like for functions, at a stationary
Second_variation
Approach to finding numerical solutions of ordinary differential equations
y {\displaystyle y} has a bounded second derivative and f {\displaystyle f} is Lipschitz continuous in its second argument, then the global truncation
Euler_method
Concept in copyright law
previously created original work (the underlying work). The derivative work becomes a second, separate work independent from the first. The transformation
Derivative_work
Generalization of the product rule in calculus
induction. If, for example, n = 2, the rule gives an expression for the second derivative of a product of two functions: ( f g ) ″ ( x ) = ∑ k = 0 2 ( 2 k )
General_Leibniz_rule
Probability distribution
{\partial }{\partial p}}\ln L(p;X)={\frac {1}{p}}-{\frac {X}{1-p}}} The second derivative of the log-likelihood function is: ∂ 2 ∂ p 2 ln L ( p ; X ) = −
Geometric_distribution
In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables
Parametric_derivative
Fundamental construction of differential calculus
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical
Generalizations of the derivative
Generalizations_of_the_derivative
Hyperbolic analogues of trigonometric functions
hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t)
Hyperbolic_functions
Formula for the derivative of a ratio of functions
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 (
Quotient_rule
Theorem in mathematics
differentiable in an open interval, with a continuous derivative, then in a neighborhood of any point where the derivative is not zero, f has an inverse function. The
Inverse_function_theorem
Time rate of change of some physical quantity of a material element in a velocity field
material derivative, including: advective derivative convective derivative derivative following the motion hydrodynamic derivative Lagrangian derivative particle
Material_derivative
Model parameters in mathematical finance
(known in calculus as partial derivatives; first-order or higher) representing the sensitivity of the price of a derivative instrument such as an option
Greeks_(finance)
Computer Language for System Modeling
calculate the second derivative of a trigonometric function, using OMShell, as a means to develop the program written below. model second_derivative Real l;
Modelica
Multivariate derivative (mathematics)
rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a
Gradient
Relationship between derivatives and integrals
each other. The second fundamental theorem says that the sum of infinitesimal changes in a quantity (the integral of the derivative of the quantity)
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Study of curves from a differential point of view
have an arc length of 1. The second derivative vectors are perpendicular to their tangent vectors. The second derivative vectors, which are the curvature
Differentiable_curve
Theorem in mathematics
there is at least one point in ( a , b ) {\displaystyle (a,b)} where the derivative equals the function's average rate of change over the whole interval.
Mean_value_theorem
Calculus of vector-valued functions
distinguished by considering the eigenvalues of the Hessian matrix of second derivatives. By Fermat's theorem, all local maxima and minima of a differentiable
Vector_calculus
Type of spline in applied mathematics
of non-smoothing spline that gives good fits to curves where the second derivative is rapidly varying. The Akima spline was published by Hiroshi Akima
Akima_spline
differentiation Stationary point Maxima and minima First derivative test Second derivative test Extreme value theorem Differential equation Differential
List_of_calculus_topics
define a new set of moments for a truncated second derivative profile starting at the second derivative minimum. If the width, σ v {\displaystyle {\sigma
Boundary_layer_thickness
Method of curve fitting
It can be proven using Rolle's theorem that if f has a continuous second derivative, then the error is bounded by | R T | ≤ ( x 1 − x 0 ) 2 8 max x 0
Linear_interpolation
Algorithm to smooth data points
curvature of the absorption band, the second derivative effectively flattens the baseline. Three measures of the derivative height, which is proportional to
Savitzky–Golay_filter
Wavelet proportional to the second derivative of a Gaussian
_{\mathbb {R} }\psi ^{2}=1\right)} second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite function. It is a special
Ricker_wavelet
Algorithm for finding zeros of functions
differentiable, its derivative is nonzero at α, and it has a second derivative at α, then the convergence is quadratic or faster. If the second derivative is not 0
Newton's_method
Set of quantities in probability theory
1 μ {\displaystyle K'(t)=(1+(e^{-t}-1)\varepsilon )^{-1}\mu } The second derivative is K ″ ( t ) = ( ε − ( ε − 1 ) e t ) − 2 μ ε e t {\displaystyle K''(t)=(\varepsilon
Cumulant
Mathematical rule for evaluating limits
functions, both of which tends to zero or infinity, by taking each function's derivative. The rule is named after the 17th-century French mathematician Guillaume
L'Hôpital's_rule
When the net force on a particle is zero
that the derivative of the function is zero at these points. To determine whether or not the system is stable or unstable, the second derivative test is
Mechanical_equilibrium
Concept in mathematical finance
the price of an output does not change linearly, but depends on the second derivative (or, loosely speaking, higher-order terms) of the modeling function
Convexity_(finance)
In mechanics, the derivative of the position vs. time graph of an object is equal to the velocity of the object. In the International System of Units,
Motion_graphs_and_derivatives
Formula for the derivative of a product
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
Product_rule
Study of mathematical algorithms for optimization problems
the second derivative or the matrix of second derivatives (called the Hessian matrix) in unconstrained problems, or the matrix of second derivatives of
Mathematical_optimization
Mathematical method in calculus
product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative
Integration_by_parts
Concept in mathematics
v = d x d t {\displaystyle v={\frac {dx}{dt}}} and expressing the second derivative of x {\displaystyle x} via the chain rule as d 2 x d t 2 = d v d t
Autonomous system (mathematics)
Autonomous_system_(mathematics)
Generalisation of the derivative of a function
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable
Weak_derivative
the second derivative of f. Similarly, the derivative of the second derivative, if it exists, is written f ′′′ and is called the third derivative of f
Glossary_of_calculus
Formula for the derivative of an inverse function
formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the
Inverse_function_rule
Feature observed in spectroscopy
also true of the third derivative; odd derivatives can be used to locate the position of a peak maximum. The second derivatives, d 2 y d x 2 {\displaystyle
Spectral_line_shape
Operation in mathematical calculus
Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals
Integral
Mathematical approximation of a function
infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum
Taylor_series
is the derivative of f with respect to this variable. The second derivative is the derivative of f ′ {\displaystyle f'} , and is denoted f ″ {\displaystyle
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Probability distribution modeling a coin toss which need not be fair
{\partial }{\partial p}}\ln L(p;X)={\frac {X}{p}}-{\frac {1-X}{1-p}}} The second derivative of the log-likelihood function is: ∂ 2 ∂ p 2 ln L ( p ; X ) = −
Bernoulli_distribution
Circulation density in a vector field
the antisymmetry in the definition of the curl, and the symmetry of second derivatives. The divergence of the curl of any vector field is equal to zero:
Curl_(mathematics)
Mathematical function defined piecewise by polynomials
continuity C2, i.e. the values and first and second derivatives are continuous. Natural means that the second derivatives of the spline polynomials are zero at
Spline_(mathematics)
Extension of cubic spline interpolation
continuous second derivative. Also, when a = − 1.0 {\displaystyle a=-1.0} , the derivative of the convolution kernel matches the derivative of the sinc
Bicubic_interpolation
Vector operator in vector calculus
exterior derivative is usually easier than working with the vector field and divergence, because unlike the divergence, the exterior derivative commutes
Divergence
Fundamental trigonometric functions
second derivative test, according to which the concavity of a function can be defined by applying the inequality of the function's second derivative greater
Sine_and_cosine
Mathematical criterion about whether a series converges
{\displaystyle f(1/n)=a_{n}} for all positive integers n and the second derivative f ″ {\displaystyle f''} exists at x = 0 {\displaystyle x=0} . Then
Convergence_tests
Specialized notation for multivariable 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 multivariate
Matrix_calculus
Approximation of a function by a polynomial
matching one derivative of f ( x ) {\textstyle f(x)} at x = a {\textstyle x=a} , this polynomial has the same first and second derivatives, as is evident
Taylor's_theorem
plane. Specifically, the Landau derivative is a second derivative of specific volume with respect to pressure. The derivative is denoted commonly using the
Landau_derivative
Description of a quantum-mechanical system
Schrödinger is nonrelativistic because it contains a first derivative in time and a second derivative in space, and therefore space and time are not on equal
Schrödinger_equation
Method to find local maxima and minima of differentiable functions on open sets
differentiable, as some stationary points are not local extrema. The second derivative, if non-zero, can be used to determine whether a local extremum at
Interior_extremum_theorem
Point where the derivative of a function is zero or undefined (in certain cases)
mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below). The value of the function
Critical_point_(mathematics)
Zero of the derivative of a function
variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing
Stationary_point
Calculus of functions of several variables
partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. A partial derivative may
Multivariable_calculus
Differential operator in mathematics
Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In
Laplace_operator
Conditions for switching order of integration in calculus
theorem – analog of Fubini's theorem for arbitrary second countable Baire spaces Symmetry of second derivatives − analogue for differentiation Fubini's nightmare –
Fubini's_theorem
SECOND DERIVATIVE
SECOND DERIVATIVE
Girl/Female
Tamil
Second
Girl/Female
Biblical
Second.
Male
English
Variant spelling of Middle English Estmond, ESMOND means "gracious protector."Â
Surname or Lastname
English
English : from an Old English personal name composed of the elements ēast ‘grace’, ‘beauty’ + mund ‘protection’. This name was also used by the Norman, among whom it represents a continental Germanic cognate of the Old English name.
Girl/Female
Muslim
Second Khalifah
Surname or Lastname
English
English : from Richward, a Norman personal name composed of the Germanic elements rīc ‘power(ful)’ + ward ‘guard’.French : from Old French record, recort ‘recollection’, ‘account’, ‘testimony’, and by extension ‘witness’, hence perhaps a nickname for someone who had given evidence in a court of law, or a metonymic occupational name for a clerk who recorded court proceedings.New England variant of French Ricard, reflecting an Americanized spelling of the Canadian pronunciation.
Female
English
From the name of the state of Arizona in the United States of America, a place considered sacred by the Native Americans. It was named after Sedona Miller Schnebly (1877-1950), the wife of the city's first postmaster. Meaning unknown.
Boy/Male
English
Protected by God. Grace and protection. From the Old English name Estmund. Commonly used as a...
Girl/Female
Tamil
Dual, Second
Boy/Male
Scottish
Second son.
Boy/Male
Scottish American
Second son.
Boy/Male
Indian
Second
Boy/Male
American, British, Christian, English, French, German
Wealthy Protector; Protected by Grace; Gracious Protector
Boy/Male
Christian & English(British/American/Australian)
Protective Grace
Female
English
Anglicized form of Scottish Gaelic Seònaid, SEONA means "God is gracious."
Boy/Male
African American American
Of man.
Girl/Female
Spanish
Lively.
Girl/Female
Indian
Second
Girl/Female
Indian
Dual, Second
Boy/Male
Scottish American Irish Russian
Second son.
SECOND DERIVATIVE
SECOND DERIVATIVE
Girl/Female
Hindu
Early morning
Surname or Lastname
English (East Midlands)
English (East Midlands) : of uncertain origin, ostensibly a patronymic, though Reaney believes it to be a nickname from Anglo-Norman French muisson ‘sparrow’.French : variant of Musset (see Mussett 1).French : nickname from Old French moisson, mousson, ‘sparrow’.French : habitational name from Mousson in Meuse-et-Moselle, named with the Latin personal name Montius + the suffix -onem, or alternatively, with Latin mons ‘mountain’ + the suffix -ionem.
Girl/Female
Greek
Alive.
Girl/Female
Bengali, Indian, Modern
Win All
Girl/Female
English
Noble maiden.
Boy/Male
Indian
Follower of Vedas; Reader of Vedas; Protecter of Vedas
Boy/Male
Gaelic
Fierce.
Male
Dutch
, man, warrior.
Boy/Male
Assamese, Hindu, Indian, Kashmiri, Punjabi, Sanskrit, Sikh
Gold
Boy/Male
Latin Russian
Attendant.
SECOND DERIVATIVE
SECOND DERIVATIVE
SECOND DERIVATIVE
SECOND DERIVATIVE
SECOND DERIVATIVE
n.
A right of inheritance belonging to a second son; a property or possession so inherited.
a.
Of the rank or degree below the best highest; inferior; second-rate; as, a second-class house; a second-class passage.
prep.
Past, out of the reach or sphere of; further than; greater than; as, the patient was beyond medical aid; beyond one's strength.
v. t.
A writing by which some act or event, or a number of acts or events, is recorded; a register; as, a record of the acts of the Hebrew kings; a record of the variations of temperature during a certain time; a family record.
v. t.
An official contemporaneous memorandum stating the proceedings of a court of justice; a judicial record.
a.
Cutting; divivding into two parts; as, a secant line.
n.
A secdond trial, experiment, or test; a second judicial trial, as of an accused person.
a.
Being of the same kind as another that has preceded; another, like a protype; as, a second Cato; a second Troy; a second deluge.
n.
The second part in a concerted piece; -- often popularly applied to the alto.
a.
To follow or attend for the purpose of assisting; to support; to back; to act as the second of; to assist; to forward; to encourage.
n.
That which is seen at a second view; a meaning beyond the literal sense; the second intention; a hidden signification.
adv.
In the second place.
n.
One who seconds or supports what another attempts, affirms, moves, or proposes; as, the seconder of an enterprise or of a motion.
a.
The sixtieth part of a minute of time or of a minute of space, that is, the second regular subdivision of the degree; as, sound moves about 1,140 English feet in a second; five minutes and ten seconds north of this place.
a.
Of the second size, rank, quality, or value; as, a second-rate ship; second-rate cloth; a second-rate champion.
a.
Having the power of second-sight.
n.
A unit for the measurement of small intervals of time, such that 1012 (ten trillion) of these units make one second.
imp. & p. p.
of Second
adv.
Secondly; in the second place.
n.
The second part in a concerted piece.