# Second Order Approximation Of First Derivative

To test your result, use it to compute the derivative of the following data in (x, f(x)) pairs at x = 8. Two Point Approximations. In this expression the ellipsis represents terms that are third order or higher. brief introduction of complex variable could be found on Ref. The slowly varying envelope approximation is often used because the resulting equations are in many cases easier to solve than the original equations, reducing the order of—all or some of—the highest-order partial derivatives. expansion g*(x) to the first or second order. These are called nite di erencestencilsand this second centered di erence is called athree point stencilfor the second derivative in one dimension. I read recently that "it is not possible to get second order accuracy at the boundaries using finite difference method, were as same is possible with finite volume method. and verifies that the approximation is accurate to first order. Restoring the consistency for functions and first-derivative approximations is fairly straight- forward and can be achieved by using correction terms as in, e. The usual method which is described in Refs. 3 Higher Order Taylor Polynomials We get better and better polynomial approximations by using more derivatives, and getting higher degreed. The second order approximation is E(Y) ≅f( )µ 1 2 + ⋅f''( )µ⋅V X( ). Derivatives Derivative Applications Limits Integrals Integral Applications Series ODE Laplace Transform Taylor/Maclaurin Series Fourier Series Statistics Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge. The value of is defined by the FDIGITS= option: If you specify the number of accurate digits using FDIGITS= r , is set to 10 - r. Complex-step approximation to the derivative In this section the complex-step approximation is shown. This new method should have the second derivative as the leading term of a non-real part. Quadratic approximation. First order formulae f0 i = 1 h (f i −f Second order formulae f0 i = 1 2h (f i+1 −f. Numerical Differentiation The derivative of a function is defined as if the limit exists • Physical examples of the derivative in action are: - Given is the position in meters of an object at time t, the first derivative with respect to t, , is the velocity in meters/second (note: The integral of velocity is position to within a constant). Example 3:. This post is part of a series of Finite Difference Method Articles. The eighth element of the opt argument specifies the type of finite-difference approximation used to compute first- or second-order derivatives and whether the finite-difference intervals, , should be computed by an algorithm of Gill et al. kkk x i 1 x i x i+1 1 -2 1 Finite Di erences October 2. The discrete operators are required to be the negative adjoints of each other. • Fi t d d i tiFirst order derivatives • Higher order derivatives • Partial differentiation • Higher order partials • Differentials • Derivatives of implicit functionsDerivatives of implicit functions • Generalized implicit function theorem • Exponential and logarithmic functions • Taylor series approximation First Order. OptFDNIPS(&nlp): Newton nonlinear interior-point method for generally constrained problems; uses second-order finite differences for Hessian approximation ; OptQNIPS(&nlp): quasi-Newton nonlinear interior-point method for generally constrained problems; uses BFGS for Hessian approximation. Neural networks are comprised of multiple hidden layers, and each of the hidden layers has. Often in my textbooks or course notes, authors and lecturers seem to take the fact that a function is slowly varying to imply that the function's second derivative is also small. We first start. The beginning of a study of functions of several variables. So we have the worst possible case for subtraction:. Objective:. Elementary derivative rules; The sine and cosine functions; The product and quotient rules; Derivatives of other trigonometric functions; The. This gives us a finite difference approximation to the derivative. Note To understand this topic, you will need to be familiar with derivatives, as discussed in Chapter 3 of Calculus Applied to the Real World. Then the exact value y(xn) is replaced by its numerical approximation yn, and the derivatives are replaced by their truncated expansions. Not sure how to go about this one, I'm assuming we use the first order Taylor approximation (tangent place) then go from there, but where to go I'm not sure. A second-order approximation using the complex-step approach is straightforward to derive; however, this approach is subject to roundoﬀ errors for small step-sizes since diﬀerence errors arise, as shown by the classic plot in Figure 1. Now we'll do something like that with the second-order approximation p 2. Our interest here is to obtain the so-called forward diﬀerence formula. We develop difference approximations for the second derivative and for the Laplacian operator. Make a plot of the first derivative at x=3 as a function of h. The FD= and FDHESSIAN= options specify the use of finite-difference approximations of the derivatives. Journal of Computational Physics 284, 687-699. One year later, he along with Thomas F Coleman developed ADMIT-1 . Since Bessel's diﬀerential equation is a second-order equation, there must be two linearly independent solutions. becomes an O((Dx)2) approximation to the second spatial derivative. snoptTL ) for information on derivative level needed. G prime prime of two times x minus two and then plus the second derivative of the function that I'm trying to approximate but the second derivative of g prime is going to be the third derivative of g. We use this fact to device a second order accurate finite difference approximation for the fractional diffusion equation. 0 2 x h h 01 2 2 2 10 0 1 12 2 Xx xx hh hh. Five Point Stencil is a generalized method to calculate zero, first and second order discrete derivative of a given time series. Second derivatives can be calculated by applying the first derivative formulas twice, or equivalently by using the central second difference formula. This will be done for. 5 and 9 is to take only the first derivatives of g(x), i. Fortunately, second order convergence can be recovered. (99) In practice, generating a ﬁnite difference approximation s tarting from the deﬁnition of the derivative can be quite tedious. 1 Simplified Solutions Based on Approximations of e z. The technical features of Julia, namely, multiple dispatch, source code via reflection, JIT compilation, and first-class access to expression parsing make implementing and using techniques from automatic differentiation. However, the closest thing I've found is numpy. Second Order Differential Equation. To finish we compare this approximation with the expressionfor a Taylor Expansion of the exact solution (going from the first line to the second we used the chain rule for partical derivatives). The FD weights at the nodes and are inx x +h for approximating the second derivative (since m = 2. First order formulae f0 i = 1 h (f i −f Second order formulae f0 i = 1 2h (f i+1 −f. Is there some reason a function couldn't have a first derivative that oscillated around zero very rapidly, for example?. First Derivative! Finite Difference Approximations! Computational Fluid Dynamics I! When using FINITE DIFFERENCE approximations, the values of f are stored at discrete points. Thus we can also estimate the first derivative, knowing two or more points; the order of the estimate depends upon the number of terms used. We will study what these look like and in particular focus on second order approximations { that is { quadratic approximations. The weights of some of the central finite difference formulas are presented in the Tables 1 and 2 for the approximations of first and second derivatives respectively. The delta-gamma approxi-mation requires the calculation of the second. (Intuitively, one can see the coeﬃcients have these values by diﬀerentiating both sides of (13) and pretending the approximation is an equality. JuliaDiff is an informal organization which aims to unify and document packages written in Julia for evaluating derivatives. A first-order or linear approximation only includes the first two terms:. First Order and Second Reliability Methods 7. Our interest here is to obtain the so-called forward diﬀerence formula. 10 Numerical Solution to First-Order Differential Equations Sofarinthischapterwehaveinvestigatedﬁrst-orderdifferentialequationsgeometrically viaslopeﬁelds,andanalyticallybytryingtoconstructexactsolutionstocertaintypesof differential equations. We use the method in the stimulation of numerical techniques for the approximate solution of linear initial value problems (IVP) in first and second order ordinary differential equations. ! h! h! f(x-h) f(x) f(x+h)! The derivatives of the function are approximated using a Taylor series! Finite Difference Approximations! Computational Fluid Dynamics I!. Geometrically, the cen-tered diﬀerence quotient represents the slope of the secant line through the two points x− h,u(x−h) and x+ h,u(x+ h) on the graph of u centered symmetrically about the point x. We will also see how tangent planes can be thought of as a linear approximation to the surface at a given point. Note that if g is twice Fréchet differentiable at x0 , then (g ′ (x0 ), 12 g ′′ (x0 )) is a second-order approximation of g at x0 , where g ′ (x0 ) and g ′′ (x0 ) are the first and second Fréchet derivatives of g at x0 , respectively. We obtain a small tip angle approximation as first order Taylor development from the first order derivatives and also develop algorithms for first and second order derivatives for this small tip angle approximation. Now consider taking h to be very small, say h = 10¡15. The FD weights at the nodes and are inx x +h for approximating the second derivative (since m = 2. Approximations of higher derivatives f00(x),f000(x),f(4)(x) etc. Neural networks are comprised of multiple hidden layers, and each of the hidden layers has. the rate of increase of acceleration, is technically known as jerk (symbol j ). Forward Diﬀerence Formula for the First Derivative We want to derive a formula that can be used to compute the ﬁrst derivative of a function at any given point. Objective:. we seek a situation where h is small compared to everything else, in order to get a good approximation to the derivative. This means that x +h and x −h are very similar in magnitude, and this means that for most f (well-behaved) that f(x +h) will be very close to f(x −h). Obtaining the derivatives of f (requires Matlab's Symbolic. The second derivative is the rate of change of the first derivative. Approximation of the first derivative by writing Taylor expansions 0 Is this how to obtain discretization for a second order derivative (backward difference, second order accurate)?. This post is part of a series of Finite Difference Method Articles. Second Derivative Test. It first does the 2nd order centered finite-difference approximation of one of the partials, and then inserts the approximation of the second partial into it (using the same formula): Inserting lines 2 and 3 into 1 gives (according to the pdf) the following:. f (x) = e-2x - x (a) Use calculus to determine the correct value of the deriv­ative at x = 2. In a similar way we can approximate the values of higher-order derivatives. The popular choices are those of order O(h2)and O(h4)and are given in Tables 6. 1 Simplified Solutions Based on Approximations of e z. This lesson is all about using the tangent line to approximate another point on our curve. There is no analogous second-order ﬁnite diﬀerence formula in the fractional case, so we will employ (see Section 4 in this paper) a Richardson extrapolation scheme in the x-direction to obtain second-order accuracy in spatial direction. In the frequency domain. As expected, one can also discretize second-order derivatives. A moving asymptotes algorithm using new local convex approximation methods with explicit solutions. and, for sufficiently small , we obtain central difference approximation of the local first order derivative by neglecting the terms of order higher than , i. The dotted line is the true derivative and the solid line is the ﬂnite dif-ference approximation. We introduced a parameter which was related to ; then two numerical schemes for variable-order Caputo fractional derivatives were derived; the second-order numerical approximation to variable-order fractional derivatives and -order approximation for are established. Interval h In this section a few examples are given on how the 1 st and 2 nd order derivatives are approximated with different order errors by various methods mentioned above. ME5286 - Lecture 6 #3 This Lecture • Examples of Edge Detection with first order derivative approximation • Edge Detection with Second order. Same meaning as second derivative meant in Cal I Derivative of the first from ECON 101 at Aachen University of Applied Sciences. 2, the second derivatives or Laplacian play an important role. The Variational Principle (Rayleigh-Ritz Approximation) Because the ground state has the lowest possible energy, we can vary a test wavefunction, minimizing the energy, to get a good estimate of the ground state energy. Is a sequel allowed to start before the end of the first book?. Approximation of the vector field using Hessian matrix, i. First, the derivative approximation of a scalar variable is summarized, followed by an extension to the second derivative. Not sure how to go about this one, I'm assuming we use the first order Taylor approximation (tangent place) then go from there, but where to go I'm not sure. Then, using our results from above,. The second derivative is the rate of change of the first derivative. where p is an integer and (1/f) (p) is the derivative of order p of the inverse of the function f. Partial derivatives. If t N and t D are constant (and assuming a constant D t), then the recurrence coefficients A and B are also constant and can be pre-computed off-line using the optimum constant-t formulas given by equations (2. Finite di erence approximations are often described in a pictorial format by giving a diagram indicating the points used in the approximation. Chapter 1 Finite Difference for Fractional Flow Equation Reading assignment: Reservoir Simulation (Mattax and Dalton 1990), Chapter 1, 2, 5, and Appendix B. If a solver requires second order information it is highly recommended that at least the first order information is given. Second Derivatives It is, of course, possible to compute higher-order derivatives of functions of two variables. October 31, 2014. 16) and a 2nd order central difference: (7. The approximations should be evaluated at the first point. 9: Different geometric interpretations of the first-order finite difference approximation related to forward, backward, and central difference approximation. Five-point endpoint formula [ ] 2nd derivative approximation (obtained by Taylor polynomial). First order of system is defined as first derivative with respect to time and second order of system is second derivative with respect to time. Also ADMAT could be used as a plug-in tool with ADMIT-1 to compute the derivative matrices for nonlinear optimization. Second or fourth-order finite difference operators, which one is most effective? Int. Dressier, J. Corollary 1: The first derivative of the shifted second kind Chebyshev polynomial is given by + = = + k n odd k n DU x k k U x ( ) 0 * * ( ) 4 ( 1) ( ) (10) B. It's a worse approximation than, say, the 2nd- or 3rd-order approximation, but it's easier to work with if accuracy isn't that important. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. Second derivatives are the same for the inner points, but they only supply the first order accurate second derivatives at the endpoints. First Derivative! Finite Difference Approximations! Computational Fluid Dynamics I! When using FINITE DIFFERENCE approximations, the values of f are stored at discrete points. Numerical Diﬀerentiation and Numerical Integration Diﬀerence and Interpolation-based Formulas for Second Derivatives (this is a second-order approximation). This means that x +h and x −h are very similar in magnitude, and this means that for most f (well-behaved) that f(x +h) will be very close to f(x −h). Second-Order Approximation to the Rotemberg Model around a Distorted Steady State* Tatiana Damjanovic† University of St Andrews Charles Nolan ‡ University of St Andrews APRIL 2010 ABSTRACT Less is known about social welfare objectives when it is costly to change prices, as in Rotemberg (1982), compared with Calvo-type models. The boundary values on the sides of the rectangle are supposed to have the sixth derivatives satisfying the Hölder condition. Formulae for Numerical Derivatives A. Second or fourth-order finite difference operators, which one is most effective? Int. (9) The continuous, second-order accurate global approximations to the curvature provide, in particular, second-order accurate curyatures half-way between pairs of mesh points on the curve. The approximation for the first and second derivatives given by equations 3 and 6 are very important in the solution of differential using the finite. To finish we compare this approximation with the expressionfor a Taylor Expansion of the exact solution (going from the first line to the second we used the chain rule for partical derivatives). For example, to find. This iteration has convergence of order (p+2). Second-Order Divided-Diﬀerence Filter Using a Generalized Complex-Step Approximation Kok-Lam Lai∗ and John L. Spline Approximation of Functions and Data This chapter introduces a number of methods for obtaining spline approximations to given functions, or more precisely, to data obtained by sampling a function. Many equations can be solved analytically using a variety of mathematical tools, but often we would like to get a computer generated approximation to the solution. This post is part of a series of Finite Difference Method Articles. a more accurate estimation of the second derivative than applying the Matlab gradient function twice. First derivative approximations A central diﬀerence approximation With second order accuracy. The end point usually occurs at the point of maximum deflection, where the absolute value of the first derivative reaches a maximum and the second derivative changes sign. MCCORMICKx SIAM J. Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function f. Neural networks are learning machines motivated by the architecture of the human brain. We first start. The approximation of first and second order pure and mixed derivative of the solution of Dirichlet problem on a rectangular parallelepiped, will be examined. These could be included but wouldn't change the conclusion. Even though the resulting solutions will only be approximations, approximate solutions can often provide a lot of insight into a. Describe the following features of Edge detection: A) Edge, Line and Spot models B) First-Order Derivative Edge Detection C) Second-Order Derivative Edge Detection Bigg Boss Check out this once. Derive first‐order and second‐order finite‐difference approximations that span across three points. The popular choices are those of order O(h2)and O(h4)and are given in Tables 6. The derivative function; Interpreting, estimating, and using the derivative; The second derivative; Limits, Continuity, and Differentiability; The Tangent Line Approximation; 2 Computing Derivatives. where p is an integer and (1/f) (p) is the derivative of order p of the inverse of the function f. First, the complex-step approximation for the first derivative of a scalar function is summarized, followed by the derivation of the second-derivative approximation. The character of H* is given by first noting that L x = f x - λp x L y = f y - λp y L λ = -(p x x + p y y - I) And thus the second derivatives are given as:. To overcome this drawback, one usually replaces the second order derivative by its finite difference. Note To understand this topic, you will need to be familiar with derivatives, as discussed in Chapter 3 of Calculus Applied to the Real World. The first step is to convert the above second-order ode into two first-order ode. Then the exact value y(xn) is replaced by its numerical approximation yn, and the derivatives are replaced by their truncated expansions. HOW TO USE THIS BOOK Stage 2 Questions in this category are for practicing skills. This is exactly the statement that the first derivative of a function is zero at the local minima. We point out that the equations. Elementary derivative rules; The sine and cosine functions; The product and quotient rules; Derivatives of other trigonometric functions; The. Higher derivatives. and verifies that the approximation is accurate to first order. First Order Differential Equations Separable Equations Homogeneous Equations Linear Equations Exact Equations Using an Integrating Factor Bernoulli Equation Riccati Equation Implicit Equations Singular Solutions Lagrange and Clairaut Equations Differential Equations of Plane Curves Orthogonal Trajectories Radioactive Decay Barometric Formula Rocket Motion Newton’s Law of Cooling Fluid Flow. If we want to approximate this to first order, it just means that you use up to the term and scrap the rest, meaning thatwhich is a first-order Taylor series approximation of about. 9: Different geometric interpretations of the first-order finite difference approximation related to forward, backward, and central difference approximation. Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function ƒ. Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants. Second or fourth-order finite difference operators, which one is most effective? Int. Then, i defined the pressure in second order. In fixed income, the result is the well-known duration-convexity approximation. A second order derivative approximation uses a point h ahead and a point h behind of the given value of x at which the second derivative of f (x) is to be found. October 31, 2014. Also ADMAT could be used as a plug-in tool with ADMIT-1 to compute the derivative matrices for nonlinear optimization. In derivatives, the result is the delta-gamma-vega-vanna approximation. Finite diﬀerence method First-order derivatives ∂u ∂x (¯x) = lim Approximation of second-order derivatives Central diﬀerence scheme. The slowly varying envelope approximation is often used because the resulting equations are in many cases easier to solve than the original equations, reducing the order of—all or some of—the highest-order partial derivatives. For example, the first derivative of sin(x) with respect to x is cos(x), and the second derivative with respect to x is -sin(x). Our main topic which we discuss here is First Order Control System. First-order approximation gx_hx. The reason for us to uti- lize first and second-order approximations is that their definitions are very simple and even discontinuous mappings may have second-order approximations. the same as the order of x as a monomial or the order of @ as a partial derivative. For example, F(x+ 2h) = X1 n=0 (2h)n n! F(n)(x); F(x 2h) = X1 n=0 ( 1)n (2h)n n! F(n)(x) (7) lead to a forward di erence approximation with second order error, F0(x) =. Neural networks are learning machines motivated by the architecture of the human brain. thus, the CFD approximation of f 00 is second-order accurate. Under these conditions a uniform approximation of order \(O(h^{4})\) (h is the grid size) is obtained for the solution of the Dirichlet problem on a square grid, its first and pure second derivatives, by a simple difference scheme. The weights of some of the central finite difference formulas are presented in the Tables 1 and 2 for the approximations of first and second derivatives respectively. Second derivative approximations. Second Derivative. then a reasonable approximation for that derivative would be to take. Second Derivatives It is, of course, possible to compute higher-order derivatives of functions of two variables. Gradient Descent: Use the first order approximation In gradient descent we only use the gradient (first order). Thus we have succeeded in writing the function f(x) in terms of its value f(a) at the point x = a, plus a linear term in (x-a) with coefficient , the first derivative at the same point x = a, with a remainder term which involves the second derivative of f(t) on the interval t = a to t = x. MCCORMICKx SIAM J. The "Second Derivative" is the derivative of the derivative of a function. De ne the Hessian H as this n n matrix of second-order partial derivatives Hf = 2 6 6 6 4 f x 1x 1 f x 1x 2 f x 1xn f x 2x. Then, approximations for multi-variable functions are presented for the Jacobian and Hessian matrices. Solution : I set us up for success with the last solved exercise, which is adapted here. Given two measurements in a pH vs. ! Friedrich Wilhelm Bessel (1784 – 1846) studied disturbances in planetary motion, which led him in 1824 to make the first systematic analysis of solutions of this equation. derivative function using a second order finite. This is illustrated by DerivativeSmoothing. The reason for us to uti- lize first and second-order approximations is that their definitions are very simple and even discontinuous mappings may have second-order approximations. Of course, to get the tangent line we do need to take derivatives, so in some way this is an application of derivatives as well. Examples: * Newtonian physics (accelaration * mass = force, acceleration is a second derivative) * Waves (the wave equation) * Hea. in the di erentiable case only at points cwhere f0(c)=0. With modern calculators and computing software it may not appear necessary to use linear approximations. VICENTEz Abstract. Higher-Order Derivatives in Engineering Applications, AD 2008, August 11 - 15 2 AD and its Applications Automatic Differentiation (AD) is a set of techniques based on the mechanical application of the chain rule to obtain derivatives of a function given as a computer program. and the approximation f(x,y)~~Q(x,y) is called the quadratic approximation to f at (a,b). 5 1 You can clearly see the eﬁect of too small an increment in the ﬂnite diﬁerence formula. Obtaining the derivatives of f (requires Matlab's Symbolic Math Toolbox) anal_deriv. Approximation of Differential Equations by Numerical Integration. We start with the Taylor expansion of the function about the point of interest, x, f(x+h) ≈ f(x)+f0(x)h+. The second-order Taylor series approximation of e x around x = 0. Derivatives- motivation Engineers often need to calculate derivatives approximately, either from data or from functions for which simple analytic forms of the derivatives don't exist. a second-order centered difference approximation: In the past, we have added additional equations representing BCs at the two exterior points. In Section 3 for the multivariate case, we will discuss a strategy to determine at each iteration a reasonably good numerical approximation to the second derivative. It first does the 2nd order centered finite-difference approximation of one of the partials, and then inserts the approximation of the second partial into it (using the same formula): Inserting lines 2 and 3 into 1 gives (according to the pdf) the following:. This is exactly the statement that the first derivative of a function is zero at the local minima. The “histogram” estimates the second derivative, because, as noted earlier, subtracting a filtered variable from the variable generates an estimate of its time derivative. To derive it from the Taylor series, simply add rather than subtract the two Taylor series of Eq. It is assume that the − ℎ order derivatives, ∈{4,5}of the functions which are given on the boundary satisfy Hölder condition. In a similar way we can approximate the values of higher-order derivatives. for approximating the second derivative (since m = 2). of the even-order derivatives. The higher the order of approximation, the more accurate the value is. Correcting second-derivative approximations, however, is more complicated. If our data is band limited, or smooth enough such the higher order derivatives are zero, then the approximation given by (4) become an exact solution. In the frequency domain. "Matlab code for Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function," QM&RBC Codes 50, Quantitative Macroeconomics & Real Business Cycles. // If the current row is the first row, then use a forward difference to compute the endpoint. Then, plus the first derivative of this thing which is the second derivative of g. snoptTL ) for information on derivative level needed. (26) The central difference approximation is accurate to and is more commonly used. There is no analogous second-order ﬁnite diﬀerence formula in the fractional case, so we will employ (see Section 4 in this paper) a Richardson extrapolation scheme in the x-direction to obtain second-order accuracy in spatial direction. Thus we see that indeed the derivative is \(c_1\) with the next term in the series of order \(h^2\). Higher derivatives. If a finite difference is divided by b − a, one gets a difference quotient. We first start. Second Derivative Test. In the second method we look for a solution of the equation in the form of the power function \(y = {x^k},\) where \(k\) is an unknown number. It is a little less well known that the third derivative, i. Spline Approximation of Functions and Data This chapter introduces a number of methods for obtaining spline approximations to given functions, or more precisely, to data obtained by sampling a function. The equation is of first orderbecause it involves only the first derivative dy dx (and not higher-order derivatives). "Matlab code for Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function," QM&RBC Codes 50, Quantitative Macroeconomics & Real Business Cycles. 2 USE OF APPROXIMATIONS OF FIRST ORDER DERIVATIVE To obtain the second order derivative at a point, one may use approximation of first. MCCORMICKx SIAM J. For the Complex Step Approximation, the first-derivative term is the leading term of the imaginary part and can be extracted without a difference operation. approximations to the second order differential are usually well behaved, however, the finite difference approximations to the first order differential may provide a greater challenge. It is possible, at least in theory, that the second-order terms all nearly cancel, so that the first-order expression is accurate but any partial second-order expansion is worse. The boundary values on the sides of the rectangle are supposed to have the sixth derivatives satisfying the Hölder condition. Thank you for your answer, but the idea is to compute approximation to first and second derivative of function (any) with *fft*. Second Order Derivative: The edge detector computed the fi rst derivative and, if it was above a certain threshold, then this results in detection of too many edge points. Second Order Approximation Of First Derivative.