Leibniz formula for derivatives

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1: 1.4 Calculus of One Variable

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§1.4(iii) Derivatives

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Higher Derivatives

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Chain Rule

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Leibniz’s Formula

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Faà Di Bruno’s Formula

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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

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§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals

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§1.5(vi) Jacobians and Change of Variables

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3: 17.2 Calculus

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§17.2(iv) Derivatives

►The $q$-derivatives of $f(z)$ are defined by … ►When $q\to 1$ the $q$-derivatives converge to the corresponding ordinary derivatives. … ►

Leibniz Rule

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4: 36.4 Bifurcation Sets

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§36.4(i) Formulas

… ► $K=2$, cusp bifurcation set: … ► $K=3$, swallowtail bifurcation set: … ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►Hyperbolic umbilic bifurcation set (codimension three): …

5: 31.12 Confluent Forms of Heun’s Equation

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31.12.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+ϵ\right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E1

Encodings:

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See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.2 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\delta }{z^2}+\frac{\gamma }{z}+1\right)\frac{dw}{dz}+\frac{\alpha z-q}{z^2}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.12.E2

Encodings:

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See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

Erratum (V1.0.7) for Equation (31.12.3)

Permalink:

http://dlmf.nist.gov/31.12.E3

Encodings:

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Errata (effective with 1.0.7):

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.4 $$\frac{d^{2}w}{{dz}^2}+\left(\gamma +z\right)z\frac{dw}{dz}+\left(\alpha z-q\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E4

Encodings:

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See also:

Annotations for §31.12, §31.12 and Ch.31

… ►For properties of the solutions of (31.12.1)–(31.12.4), including connection formulas, see Bühring (1994), Ronveaux (1995, Parts B,C,D,E), Wolf (1998), Lay and Slavyanov (1998), and Slavyanov and Lay (2000). …

6: 1.2 Elementary Algebra

… ►and $f^{(k)}$ is the $k$-th derivative of $f$ (§1.4(iii)). … ►where $det(𝐀)$ is defined by the Leibniz formula … ►Formula (1.2.77) is more generally valid for all square matrices $𝐀$, not necessarily non-defective, see Hall (2015, Thm 2.12).

7: 4.24 Inverse Trigonometric Functions: Further Properties

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§4.24(ii) Derivatives

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4.24.7 $$\frac{d}{dz}\arcsin z={(1-z^2)}^{-1/2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.52

Permalink:

http://dlmf.nist.gov/4.24.E7

Encodings:

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See also:

Annotations for §4.24(ii), §4.24 and Ch.4

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4.24.8 $$\frac{d}{dz}\arccos z=-{(1-z^2)}^{-1/2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arccos z$: arccosine function and $z$: complex variable

A&S Ref:

4.4.53

Permalink:

http://dlmf.nist.gov/4.24.E8

Encodings:

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See also:

Annotations for §4.24(ii), §4.24 and Ch.4

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4.24.9 $$\frac{d}{dz}\arctan z=\frac{1}{1+z^2}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.54

Permalink:

http://dlmf.nist.gov/4.24.E9

Encodings:

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See also:

Annotations for §4.24(ii), §4.24 and Ch.4

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§4.24(iii) Addition Formulas

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8: 3.4 Differentiation

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Two-Point Formula

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Three-Point Formula

… ►For corresponding formulas for second, third, and fourth derivatives, with $t=0$, see Collatz (1960, Table III, pp. 538–539). For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). … ►Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives. …

9: 31.18 Methods of Computation

… ►Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of $z$; see Laĭ (1994) and Lay et al. (1998). …

10: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

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§13.3(ii) Differentiation Formulas

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13.3.22 $$\frac{d}{dz}U(a,b,z)=-aU(a+1,b+1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

13.4.21

Permalink:

http://dlmf.nist.gov/13.3.E22

Encodings:

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See also:

Annotations for §13.3(ii), §13.3 and Ch.13

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13.3.23 $$\frac{d^n}{{dz}^n}U(a,b,z)={(-1)}^n{\left(a\right)}_{n}U(a+n,b+n,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.4.22

Referenced by:

§13.3(i)

Permalink:

http://dlmf.nist.gov/13.3.E23

Encodings:

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See also:

Annotations for §13.3(ii), §13.3 and Ch.13

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13.3.29 $${\left(z\frac{d}{dz}z\right)}^n=z^n\frac{d^n}{{dz}^n}{z}^n,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.15(ii), §13.3(ii)

Permalink:

http://dlmf.nist.gov/13.3.E29

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See also:

Annotations for §13.3(ii), §13.3 and Ch.13