# differentiation

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##### 1: 27.20 Methods of Computation: Other Number-Theoretic Functions
A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau\left(n\right)$, and the values can be checked by the congruence (27.14.20). …
##### 2: 3.4 Differentiation
###### §3.4 Differentiation
and follows from the differentiated form of (3.3.4). …
##### 3: 2.1 Definitions and Elementary Properties
###### §2.1(ii) Integration and Differentiation
Differentiation requires extra conditions. …This result also holds with both $O$’s replaced by $o$’s. … means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left((x-c)^{n}\right)$ as $x\to c$ in $\mathbf{X}$. … Differentiation, however, requires the kind of extra conditions needed for the $O$ symbol (§2.1(ii)). …
##### 5: 9.10 Integrals
9.10.1 $\int_{z}^{\infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi\left(% \operatorname{Ai}\left(z\right)\operatorname{Gi}'\left(z\right)-\operatorname{% Ai}'\left(z\right)\operatorname{Gi}\left(z\right)\right),$
9.10.8 $\int zw(z)\,\mathrm{d}z=w^{\prime}(z),$
9.10.9 $\int z^{2}w(z)\,\mathrm{d}z=zw^{\prime}(z)-w(z),$
9.10.10 $\int z^{n+3}w(z)\,\mathrm{d}z=z^{n+2}w^{\prime}(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)% \int z^{n}w(z)\,\mathrm{d}z,$ $n=0,1,2,\ldots.$
9.10.20 $\int_{0}^{x}\!\!\int_{0}^{v}\operatorname{Ai}\left(t\right)\,\mathrm{d}t\,% \mathrm{d}v=x\int_{0}^{x}\operatorname{Ai}\left(t\right)\,\mathrm{d}t-% \operatorname{Ai}'\left(x\right)+\operatorname{Ai}'\left(0\right),$
##### 6: 1.5 Calculus of Two or More Variables
$\frac{\,{\partial}^{2}f}{\,\partial y\,\partial x}=\frac{\partial}{\partial y}% \left(\frac{\partial f}{\partial x}\right).$
###### §1.5(iv) Leibniz’s Theorem for Differentiation of Integrals
1.5.24 $\frac{\mathrm{d}}{\mathrm{d}x}\int^{d}_{c}f(x,y)\,\mathrm{d}y=\int^{d}_{c}% \frac{\partial f}{\partial x}\,\mathrm{d}y,$ $a.
##### 7: 1.8 Fourier Series
###### §1.8(iii) Integration and Differentiation
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 $f^{\prime}(x)$. …