18. Finite Differences - First Derivative#
18.1. Forward differencing#
\[\begin{split}
\begin{aligned}
f\left(x_0+h\right)&=
f\left(x_0\right)+
hf^{\prime}\left(x_0\right)+
\frac{h^2}{2!}f^{\prime\prime}\left(x_0\right)+
\cdots+
\frac{h^n}{n!}f^{\left(n\right)}\left(x_0\right)+
\cdots\\
f^{\prime}\left(x_0\right)&=
\dfrac{f\left(x_0+h\right)-f\left(x_0\right)}{h}-
\left(
\dfrac{h}{2!}
f^{\prime\prime}\left(x_0\right)+
\dfrac{h^{2}}{3!}
f^{\prime\prime\prime}\left(x_0\right)+
\cdots+
\dfrac{h^{n-1}}{n!}
f^{\left(n\right)}\left(x_0\right)+
\cdots
\right)
\\
f^{\prime}\left(x_0\right)&=
\dfrac{f\left(x_0+h\right)-f\left(x_0\right)}{h}+
\mathcal{O}\left(h\right)
\end{aligned}
\end{split}\]
18.2. Backward differencing#
\[\begin{split}
\begin{aligned}
f\left(x_0-h\right)&=
f\left(x_0\right)-
hf^{\prime}\left(x_0\right)+
\frac{h^2}{2!}f^{\prime\prime}\left(x_0\right)-
\cdots+
{\left(-1\right)}^{n}
\frac{h^n}{n!}f^{\left(n\right)}\left(x_0\right)+
\cdots\\
f^{\prime}\left(x_0\right)&=
\dfrac{f\left(x_0\right)-f\left(x_0-h\right)}{h}+
\left(
\dfrac{h}{2!}
f^{\prime\prime}\left(x_0\right)+
\dfrac{h^{2}}{3!}
f^{\prime\prime\prime}\left(x_0\right)+
\cdots+
{\left(-1\right)}^{n}
\dfrac{h^{n-1}}{n!}
f^{\left(n\right)}\left(x_0\right)+
\cdots
\right)\\
f^{\prime}\left(x_0\right)&=
\dfrac{f\left(x_0\right)-f\left(x_0-h\right)}{h}+
\mathcal{O}\left(h\right)
\end{aligned}
\end{split}\]
18.3. Arriving at Central difference#
\[\begin{split}
\begin{aligned}
f\left(x_0+h\right)&=
f\left(x_0\right)+
hf^{\prime}\left(x_0\right)+
\frac{h^2}{2!}f^{\prime\prime}\left(x_0\right)+
\cdots+
\frac{h^n}{n!}f^{\left(n\right)}\left(x_0\right)+
\cdots\\
f\left(x_0-h\right)&=
f\left(x_0\right)-
hf^{\prime}\left(x_0\right)+
\frac{h^2}{2!}f^{\prime\prime}\left(x_0\right)-
\cdots+
{\left(-1\right)}^{n}
\frac{h^n}{n!}f^{\left(n\right)}\left(x_0\right)+
\cdots\\
f^{\prime}\left(x_0\right)&=
\dfrac{f\left(x_0+h\right)-f\left(x_0-h\right)}{2h}-
\left(
\dfrac{h^{2}}{3!}
f^{\prime\prime\prime}\left(x_0\right)+
\cdots+
\dfrac{h^{n-1}}{n!}
f^{\left(n\right)}\left(x_0\right)+
\cdots
\right)\\
f^{\prime}\left(x_0\right)&=
\dfrac{f\left(x_0+h\right)-f\left(x_0-h\right)}{2h}+
\mathcal{O}\left(h^{2}\right)
\end{aligned}
\end{split}\]
19. Polynomial derivative estimation#
import matplotlib.pyplot as plt
import numpy as np
from numpy.polynomial import Polynomial
p = Polynomial(coef=[9, -3, 7, -4])
p_prime = p.deriv()
p_prime(0)
np.float64(-3.0)
plt.plot(*p.linspace(), lw=0.5);
