# derivatives

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##### 1: 4.20 Derivatives and Differential Equations
###### §4.20 Derivatives and Differential Equations
4.20.9 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w=0,$
##### 2: 4.34 Derivatives and Differential Equations
###### §4.34 Derivatives and Differential Equations
4.34.7 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w=0,$
4.34.8 $\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=1,$
4.34.9 $\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=-1,$
4.34.10 $\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}=1,$
##### 3: 4.7 Derivatives and Differential Equations
###### §4.7(i) Logarithms
4.7.5 $\frac{\mathrm{d}w}{\mathrm{d}z}=\frac{f^{\prime}(z)}{f(z)}$
##### 4: 36.10 Differential Equations
###### §36.10(ii) Partial Derivatives with Respect to the $x_{n}$
36.10.7 $\frac{{\partial}^{2n}\Psi_{2}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{2}}{{\partial y}^{n}}.$
36.10.8 $\frac{{\partial}^{2n}\Psi_{3}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{3}}{{\partial y}^{n}},$
36.10.10 $\frac{{\partial}^{3n}\Psi_{3}}{{\partial y}^{3n}}=i^{n}\frac{{\partial}^{2n}% \Psi_{3}}{{\partial z}^{2n}}.$
##### 5: 19.18 Derivatives and Differential Equations
###### §19.18(i) Derivatives
Let $\partial_{j}=\ifrac{\partial}{\partial z_{j}}$, and $\mathbf{e}_{j}$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. …
19.18.14 $\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{\partial}^{2}w}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial w}{\partial y}.$
19.18.15 $\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{\partial}^{2}W}{{\partial x}^{% 2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}.$
##### 6: 10.38 Derivatives with Respect to Order
###### §10.38 Derivatives with Respect to Order
10.38.2 $\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}=\tfrac{1}{2}\pi\csc\left(% \nu\pi\right)\*\left(\frac{\partial I_{-\nu}\left(z\right)}{\partial\nu}-\frac% {\partial I_{\nu}\left(z\right)}{\partial\nu}\right)-\pi\cot\left(\nu\pi\right% )K_{\nu}\left(z\right),$ $\nu\notin\mathbb{Z}$.
For $\ifrac{\partial I_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.38.1), (10.38.2), and (10.38.4). …
$\left.\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=-K_{0}% \left(z\right),$
$\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=0.$
##### 7: 7.10 Derivatives
###### §7.10 Derivatives
7.10.1 $\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{\mathrm{d}z}^{n+1}}=(-1)^{n}% \frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}},$ $n=0,1,2,\dots$.
$\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{d}z}=-\pi z\mathrm{g}\left(z% \right),$
$\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{d}z}=\pi z\mathrm{f}\left(z% \right)-1.$
##### 8: 1.5 Calculus of Two or More Variables
###### §1.5(i) Partial Derivatives
The function $f(x,y)$ is continuously differentiable if $f$, $\ifrac{\partial f}{\partial x}$, and $\ifrac{\partial f}{\partial y}$ are continuous, and twice-continuously differentiable if also $\ifrac{{\partial}^{2}f}{{\partial x}^{2}}$, $\ifrac{{\partial}^{2}f}{{\partial y}^{2}}$, ${\partial}^{2}f/\partial x\partial y$, and ${\partial}^{2}f/\partial y\partial x$ are continuous. …
###### Chain Rule
Suppose that $a,b,c$ are finite, $d$ is finite or $+\infty$, and $f(x,y)$, $\ifrac{\partial f}{\partial x}$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times[c,d)$. …
##### 9: 12.17 Physical Applications
12.17.2 $\nabla^{2}=\frac{{\partial}^{2}}{{\partial x}^{2}}+\frac{{\partial}^{2}}{{% \partial y}^{2}}+\frac{{\partial}^{2}}{{\partial z}^{2}}$
12.17.4 $\frac{1}{\xi^{2}+\eta^{2}}\left(\frac{{\partial}^{2}w}{{\partial\xi}^{2}}+% \frac{{\partial}^{2}w}{{\partial\eta}^{2}}\right)+\frac{{\partial}^{2}w}{{% \partial\zeta}^{2}}+k^{2}w=0.$
$\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}\xi}^{2}}+\left(\sigma\xi^{2}+\lambda% \right)U=0,$
$\frac{{\mathrm{d}}^{2}V}{{\mathrm{d}\eta}^{2}}+\left(\sigma\eta^{2}-\lambda% \right)V=0,$
$\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\zeta}^{2}}+\left(k^{2}-\sigma\right)W=0,$
##### 10: 22.13 Derivatives and Differential Equations
###### §22.13(i) Derivatives
Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. …
22.13.1 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sn}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{sn}^{2}}\left(z,k\right)\right)\left(1-k^{2}{% \operatorname{sn}^{2}}\left(z,k\right)\right),$