differentiation

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11: Philip J. Davis

… ►Davis also co-authored a second Chapter, “Numerical Interpolation, Differentiation, and Integration” with Ivan Polonsky. …

12: 1.5 Calculus of Two or More Variables

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${\displaystyle \frac{{\partial }^{2}f}{\partial y\partial x}}={\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{\partial f}{\partial x}}\right).$

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

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1.5.24 $$\frac{d}{dx}{\int }_c^{d}f(x,y)dy={\int }_c^d\frac{\partial f}{\partial x}dy,$$ .

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

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http://dlmf.nist.gov/1.5.E24

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Annotations for §1.5(iv), §1.5(iv), §1.5 and Ch.1

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13: 4.40 Integrals

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14: 5.15 Polygamma Functions

… ►Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. …

15: 5.19 Mathematical Applications

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16: 10.12 Generating Function and Associated Series

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17: 1.9 Calculus of a Complex Variable

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§1.9(ii) Continuity, Point Sets, and Differentiation

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Differentiation

… ►Differentiability automatically implies continuity. ►

Cauchy–Riemann Equations

… ►Lastly, a power series can be differentiated any number of times within its circle of convergence: …

18: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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19: 4.26 Integrals

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20: 7.7 Integral Representations

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