derivative of the Bessel function, $J'_0$ (see [@Abramovitz §9.1.28]): $$\label{derivative_J_0++} J'_0(x)\approx H'_n(x)=
-\frac{1}{n}\sum_{k=1}^{n}
 \cos \tfrac{(2k-1)\pi}{4n}\, \sin(x\cdot\cos \tfrac{(2k-1)\pi}{4n})$$ with the error (see (\[Gauss-Chebyshev-error\])) $$\label{derivative_J_0_super_error} |J'_0(x)-H'_n(x)|{\leqslant}\frac{|x|^{4n-1}}{2^{4n-1}(4n-1)!}.$$

For comparison, look at the approximant from [@YF Table 2 and Formula (30)]: $$\label{YF_J_1}
 J'_0(x)\approx \Omega_{13}(x):=-\sum_{k=1}^{13}\alpha_m \sin(\beta_m
x),\quad \alpha_k\in(0.001,0.08),\quad \beta_k\in(0.12,1).$$ Calculations show that the formula (\[derivative\_J\_0++\]) with the same $n=13$ has noticeably less error in the segment $[0,40]$: $$\label{YF_J_0_cos++++}
 {\rm log}_{10}\frac{|J'_0(x)-\Omega_{13}(x)|}{|J'_0(x)-H'_{13}(x)|}>
 M_1(x):=\frac{40}{x+1},
\qquad x\in [0,40].$$ Moreover, in the segment $[0,10]$ the minorant $M_1(x)$ can be exchanged by the more precise $ \tilde M_1(x)=43\log_{10}
\frac{1}{x}+48\to\infty$, $x\to 0$. Note that the absolute error of (\[YF\_J\_1\]) in $[0,10]$ is close to $\varepsilon=10^{-12}$.

For $40{\leqslant}x{\leqslant}45$ the absolute error of the formula (\[derivative\_J\_0++\]) is also less than the one of (\[YF\_J\_1\]). For $x{\geqslant}45$ the absolute error of both formulas can be $10^{-2}$ and thus use of them may be unreasonable.

Let us obtain one more representation for the function $J_0$ by taking $$\label{sinc}
\textrm{sinc}\; x=\frac{\sin x}{x}=\sum_{m=0}^\infty
\frac{(-1)^m}{(2m+1)!}x^{2m}$$ as a basis function and using the approach for interpolation of even functions from Remark \[remark\_even\_odd\]. Namely, we take into account that $J_0$ and $\textrm{sinc}$ are even and interpolate the function $\tilde{f}(t):=J_0(x)$, $t=x^2$, by amplitude and frequency sums with the basis function ${h}(t)=\textrm{sinc}\; x$. 