Calcular las derivadas en los puntos \( x = [1, 4, 6, 7, 11] \) de $$ \begin{equation*} f(x) = \ln{x} \end{equation*} $$
Derivada cero $$ \begin{align*} f[x_{0}] &= f(x_{0}) = f(1) = 0 \\ f[x_{1}] &= f(x_{1}) = f(4) = 1.386294 \\ f[x_{2}] &= f(x_{2}) = f(6) = 1.791759 \\ f[x_{3}] &= f(x_{3}) = f(7) = 1.945910 \\ f[x_{4}] &= f(x_{4}) = f(11) = 2.397895 \end{align*} $$
Primera derivada $$ \begin{align*} f[x_{0}, x_{1}] &= \frac{f[x_{1}] - f[x_{0}]}{x_{1} - x_{0}} = \frac{1.386294 - 0}{4 - 1} = 0.462098 \\ f[x_{1}, x_{2}] &= \frac{f[x_{2}] - f[x_{1}]}{x_{2} - x_{1}} = \frac{1.791759 - 1.386294}{6 - 4} = 0.202732 \\ f[x_{2}, x_{3}] &= \frac{f[x_{3}] - f[x_{2}]}{x_{3} - x_{2}} = \frac{1.945910 - 1.791759}{7 - 6} = 0.154151 \\ f[x_{3}, x_{4}] &= \frac{f[x_{4}] - f[x_{3}]}{x_{4} - x_{3}} = \frac{2.397895 - 1.945910}{11 - 7} = 0.112996 \end{align*} $$
Segunda derivada $$ \begin{align*} f[x_{0}, x_{1}, x_{2}] &= \frac{f[x_{1}, x_{2}] - f[x_{0}, x_{1}]}{x_{2} - x_{0}} = \frac{0.202732 - 0.462098}{6 - 1} = -0.051873 \\ f[x_{1}, x_{2}, x_{3}] &= \frac{f[x_{2}, x_{3}] - f[x_{1}, x_{2}]}{x_{3} - x_{1}} = \frac{0.154151 - 0.202732}{7 - 4} = -0.016194 \\ f[x_{2}, x_{3}, x_{4}] &= \frac{f[x_{3}, x_{4}] - f[x_{2}, x_{3}]}{x_{4} - x_{2}} = \frac{0.112996 - 0.154151}{11 - 6} = -0.008231 \end{align*} $$
Tercera derivada $$ \begin{align*} f[x_{0}, x_{1}, x_{2}, x_{3}] &= \frac{f[x_{1}, x_{2}, x_{3}] - f[x_{0}, x_{1}, x_{2}]}{x_{3} - x_{0}} = \frac{-0.016194 - (-0.051873)}{7 - 1} = 0.005946 \\ f[x_{1}, x_{2}, x_{3}, x_{4}] &= \frac{f[x_{2}, x_{3}, x_{4}] - f[x_{1}, x_{2}, x_{3}]}{x_{4} - x_{1}} = \frac{-0.008231 - (-0.016194)}{11 - 4} = 0.001138 \end{align*} $$
Cuarta derivada $$ \begin{equation*} f[x_{0}, x_{1}, x_{2}, x_{3}, x_{4}] = \frac{f[x_{1}, x_{2}, x_{3}, x_{4}] - f[x_{0}, x_{1}, x_{2}, x_{3}]}{x_{4} - x_{0}} = \frac{0.00138 - 0.005946}{11 - 1} = -0.000481 \end{equation*} $$
Resumiendo en una tabla $$ \begin{equation*} \begin{array}{c|ccccc} x & f & f' & f'' & f''' & f^{(4)} \\ \hline 1 & 0 & & & & \\ & & 0.462098 & & & \\ 4 & 1.386294 & & -0.051873 & & \\ & & 0.202732 & & 0.005946 & \\ 6 & 1.791759 & & -0.016194 & & -0.000481 \\ & & 0.154151 & & 0.001138 & \\ 7 & 1.945910 & & -0.008231 & & \\ & & 0.112996 & & & \\ 11 & 2.397895 & & & & \end{array} \end{equation*} $$