# L?H�pital rule for derivatives

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## 1—10 of 292 matching pages

##### 1: 3.8 Nonlinear Equations

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###### §3.8(ii) Newton’s Rule

… ► … ►Newton’s rule is given by … ►Another iterative method is*Halley’s rule*: …The rule converges locally and is cubically convergent. …##### 2: 1.5 Calculus of Two or More Variables

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###### §1.5(i) Partial Derivatives

… ►The function $f(x,y)$ is*continuously differentiable*if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous,*and twice-continuously differentiable*if also ${\partial}^{2}f/{\partial x}^{2}$, ${\partial}^{2}f/{\partial y}^{2}$, ${\partial}^{2}f/\partial x\partial y$, and ${\partial}^{2}f/\partial y\partial x$ are continuous. … ►###### Chain Rule

… ►Suppose that $a,b,c$ are finite, $d$ is finite or $+\mathrm{\infty}$, and $f(x,y)$, $\partial f/\partial x$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times [c,d)$. … ►###### §1.5(vi) Jacobians and Change of Variables

…##### 3: 18.40 Methods of Computation

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►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in $\mu (x)$, using what is referred to as the

*Stieltjes derivative*which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12). ►###### Derivative Rule Approach

►An alternate, and highly efficient, approach follows from the*derivative rule conjecture*, see Yamani and Reinhardt (1975), and references therein, namely that … ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. … ►Further,*exponential*convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty},\mathrm{\infty})$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …##### 4: 3.5 Quadrature

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###### §3.5(i) Trapezoidal Rules

… ►The*composite trapezoidal rule*is … ►###### §3.5(ii) Simpson’s Rule

… ►###### §3.5(iv) Interpolatory Quadrature Rules

… ► …##### 5: 9.17 Methods of Computation

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►The trapezoidal rule (§3.5(i)) is then applied.
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►In consequence of §9.6(i), algorithms for generating Bessel functions, Hankel functions, and modified Bessel functions (§10.74) can also be applied to $\mathrm{Ai}\left(z\right)$, $\mathrm{Bi}\left(z\right)$, and their derivatives.
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►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations.
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►For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).

##### 6: 1.6 Vectors and Vector-Valued Functions

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►where $\mathbf{n}$ is the unit vector normal to $\mathbf{a}$ and $\mathbf{b}$ whose direction is determined by the right-hand rule; see Figure 1.6.1.
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1.6.19
$$\nabla =\mathbf{i}\frac{\partial}{\partial x}+\mathbf{j}\frac{\partial}{\partial y}+\mathbf{k}\frac{\partial}{\partial z}.$$

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►where $\partial g/\partial n=\nabla g\cdot \mathbf{n}$ is the derivative of $g$ normal to the surface outwards from $V$ and $\mathbf{n}$ is the unit outer normal vector.
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##### 7: 1.4 Calculus of One Variable

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###### §1.4(iii) Derivatives

… ►###### Higher Derivatives

… ►###### Chain Rule

… ►###### Leibniz’s Formula

… ►###### L’Hôpital’s Rule

…##### 8: 10.74 Methods of Computation

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###### §10.74(vi) Zeros and Associated Values

►Newton’s rule (§3.8(i)) or Halley’s rule (§3.8(v)) can be used to compute to arbitrarily high accuracy the real or complex zeros of all the functions treated in this chapter. Necessary values of the first derivatives of the functions are obtained by the use of (10.6.2), for example. Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. … ► …##### 9: Bibliography R

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High precision Chebyshev expansions for Airy functions and their derivatives.
Technical report
University of Birmingham Computer Centre.
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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (4), pp. 91.
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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (3), pp. 56–64.
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Finite-sum rules for Macdonald’s functions and Hankel’s symbols.
Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
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