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1: 18.32 OP’s with Respect to Freud Weights

… ►where $Q(x)$ is real, even, nonnegative, and continuously differentiable. …

2: 1.4 Calculus of One Variable

… ►When this limit exists $f$ is differentiable at $x$. … ►When $n\ge 1$, $f$ is continuously differentiable on $I$. … … ►

Mean Value Theorem

… ►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …

3: 1.6 Vectors and Vector-Valued Functions

… ►when $f$ is continuously differentiable. … ►The curve $C$ is piecewise differentiable if $𝐜$ is piecewise differentiable. … … ►For $x$, $y$, and $z$ continuously differentiable, the vectors … ►For $f$ and $g$ twice-continuously differentiable functions …

4: 4.12 Generalized Logarithms and Exponentials

… ►Both $\varphi (x)$ and $\psi (x)$ are continuously differentiable. … ►For $C^{\mathrm{\infty }}$ generalized logarithms, see Walker (1991). …

5: 2.3 Integrals of a Real Variable

… ►converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. … ►If, in addition, $q(t)$ is infinitely differentiable on $[0,\mathrm{\infty })$ and … ►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. … ►Assume that $q(t)$ again has the expansion (2.3.7) and this expansion is infinitely differentiable, $q(t)$ is infinitely differentiable on $(0,\mathrm{\infty })$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots }$, converges at $t=\mathrm{\infty }$, uniformly for all sufficiently large $x$. … ►

On $(a,b)$, $p(t)$ and $q(t)$ are infinitely differentiable and $p^{\prime }(t)>0$.

…

6: 1.13 Differential Equations

… ►Let $W(z)$ satisfy (1.13.14), $\zeta (z)$ be any thrice-differentiable function of $z$, and ►

1.13.18 $$U(z)={({\zeta }^{\prime }(z))}^{1/2}W(z).$$

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Symbols:

$z$: variable, $W(\xi )$: solution, $\zeta (z)$: thrice-differentiable function and $U(z)$: solution

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http://dlmf.nist.gov/1.13.E18

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Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

… ►

1.13.19 $$\frac{d^{2}U}{{d\zeta }^2}=\left({\dot{z}}^{2}H(z)-\frac{1}{2}\{z,\zeta \}\right)U.$$

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Symbols:

$\{z,\zeta \}$: Schwarzian derivative, $\frac{df}{dx}$: derivative, $z$: variable, $H(z)$ , $\zeta (z)$: thrice-differentiable function and $U(z)$: solution

Permalink:

http://dlmf.nist.gov/1.13.E19

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Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

… ►

1.13.21 $$\{z,\zeta \}={(d\xi /d\zeta )}^2\{z,\xi \}+\{\xi ,\zeta \}.$$

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Symbols:

$\{z,\zeta \}$: Schwarzian derivative, $\frac{df}{dx}$: derivative, $z$: variable, $\xi$: change of variable and $\zeta (z)$: thrice-differentiable function

Permalink:

http://dlmf.nist.gov/1.13.E21

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Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

►

1.13.22 $$\{z,\zeta \}=-{(dz/d\zeta )}^2\{\zeta ,z\}.$$

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Symbols:

$\{z,\zeta \}$: Schwarzian derivative, $\frac{df}{dx}$: derivative, $z$: variable and $\zeta (z)$: thrice-differentiable function

Permalink:

http://dlmf.nist.gov/1.13.E22

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Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

…

7: 1.5 Calculus of Two or More Variables

… ►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. … ►

1.5.6 $$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}.$$

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Symbols:

$\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/1.5.E6

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

… ►If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\partial F/\partial y\ne 0$ at $(a,b)$, then in a neighborhood of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and $g^{\prime }(x)=-F_x/F_y$. … ►If $f$ is $n+1$ times continuously differentiable, then … ►Sufficient conditions for validity are: (a) $f$ and $\partial f/\partial x$ are continuous on a rectangle $a\le x\le b$, $c\le y\le d$; (b) when $x\in [a,b]$ both $\alpha (x)$ and $\beta (x)$ are continuously differentiable and lie in $[c,d]$. …

8: 2.8 Differential Equations with a Parameter

… ►The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to $\xi$. … ►Again, $u>0$ and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$. … ►The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to $\xi$. …

9: 18.38 Mathematical Applications

… ►In consequence, expansions of functions that are infinitely differentiable on $[-1,1]$ in series of Chebyshev polynomials usually converge extremely rapidly. …

10: 1.8 Fourier Series

… ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. … ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $|f^{\prime \prime }(x)|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. …