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1: 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. …

2: 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 …

3: 4.12 Generalized Logarithms and Exponentials

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

4: 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 $p^{\prime }(t)>0$.

…

5: 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

Permalink:

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 of $f$ with respect to $x$ , $z$: variable, $H(z)$ , $\zeta (z)$: thrice-differentiable function, $U(z)$: solution and $\dot{z}$: derivative of $z$ with respect to $\zeta$

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

… ►

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

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Symbols:

$\{z,\zeta \}$: Schwarzian derivative, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: 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

… ►As the interval $[a,b]$ is mapped, one-to-one, onto $[0,c]$ by the above definition of $t$, the integrand being positive, the inverse of this same transformation allows $\widehat{q}(t)$ to be calculated from $p,q,\rho$ in (1.13.31), $p,\rho \in C^2(a,b)$ and $q\in C(a,b)$. …

6: 18.32 OP’s with Respect to Freud Weights

… ►where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $x{Q}^{\prime }(x)$ increases for $x>0$, and $Q^{\prime }(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$, see Freud (1969). …

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 of $x$

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: 1.9 Calculus of a Complex Variable

… ►

Differentiation

►A function $f(z)$ is complex differentiable at a point $z$ if the following limit exists: … ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. … ►A function $f(z)$ is said to be analytic (holomorphic) at $z=z_0$ if it is complex differentiable in a neighborhood of $z_0$. … ►An arc $C$ is given by $z(t)=x(t)+\mathrm{i}y(t)$, $a\le t\le b$, where $x$ and $y$ are continuously differentiable. …

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 $\left|f^{\prime \prime }(x)\right|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. …