Polinomio de Lagrange

$$ \begin{equation} f_n(x) = \sum_{i=0}^{n} \ell_{i} f(x_{i}) \tag{1} \end{equation} $$

Polinomio lineal

$$ \begin{equation*} f(x) = \frac{x - x_{1}}{x_{0} - x_{1}} f(x_{0}) + \frac{x - x_{0}}{x_{1} - x_{0}} f(x_{1}) \end{equation*} $$

Primera derivada $$ \begin{equation} f'(x) = \frac{1}{x_{0} - x_{1}} f(x_{0}) + \frac{1}{x_{1} - x_{0}} f(x_{1}) \tag{2} \end{equation} $$

Polinomio cuadrático

$$ \begin{equation*} f(x) = \frac{(x - x_{1})(x - x_{2})}{(x_{0} - x_{1})(x_{0} - x_{2})} f(x_{0}) + \frac{(x - x_{0})(x - x_{2})}{(x_{1} - x_{0})(x_{1} - x_{2})} f(x_{1}) + \frac{(x - x_{0})(x - x_{1})}{(x_{2} - x_{0})(x_{2} - x_{1})} f(x_{2}) \end{equation*} $$

Primera derivada $$ \begin{equation} f'(x) = \frac{2 x - (x_{1} - x_{2})}{(x_{0} - x_{1})(x_{0} - x_{2})} f(x_{0}) - \frac{2 x - (x_{0} - x_{2})}{(x_{1} - x_{0})(x_{1} - x_{2})} f(x_{1}) + \frac{2 x - (x_{0} - x_{1})}{(x_{2} - x_{0})(x_{2} - x_{1})} f(x_{2}) \tag{3} \end{equation} $$

Segunda derivada $$ \begin{equation} f''(x) = \frac{2}{(x_{0} - x_{1})(x_{0} - x_{2})} f(x_{0}) - \frac{2}{(x_{1} - x_{0})(x_{1} - x_{2})} f(x_{1}) + \frac{2}{(x_{2} - x_{0})(x_{2} - x_{1})} f(x_{2}) \tag{4} \end{equation} $$

Polinomio cúbico

$$ \begin{equation*} f(x) = \frac{(x - x_{1})(x - x_{2})(x - x_{3})}{(x_{0} - x_{1})(x_{0} - x_{2})(x_{0} - x_{3})} f(x_{0}) + \frac{(x - x_{0})(x - x_{2})(x - x_{3})}{(x_{1} - x_{0})(x_{1} - x_{2})(x_{1} - x_{3})} f(x_{1}) + \frac{(x - x_{0})(x - x_{1})(x - x_{3})}{(x_{2} - x_{0})(x_{2} - x_{1})(x_{2} - x_{3})} f(x_{2}) + \frac{(x - x_{0})(x - x_{1})(x - x_{2})}{(x_{3} - x_{0})(x_{3} - x_{1})(x_{3} - x_{2})} f(x_{3}) \end{equation*} $$

Primera derivada $$ \begin{equation*} f'(x) = \frac{3 x^{2} - 2 x (x_{1} + x_{2} + x_{3}) + x_{1} (x_{2} + x_{3}) + x_{2} x_{3}}{(x_{0} - x_{1})(x_{0} - x_{2})(x_{0} - x_{3})} f(x_{0}) + \frac{3 x^{2} - 2 x (x_{0} + x_{2} + x_{3}) + x_{0} (x_{2} + x_{3}) + x_{2} x_{3}}{(x_{1} - x_{0})(x_{1} - x_{2})(x_{1} - x_{3})} f(x_{1}) + \frac{3 x^{2} - 2 x (x_{0} + x_{1} + x_{3}) + x_{0} (x_{1} + x_{3}) + x_{1} x_{3}}{(x_{2} - x_{0})(x_{2} - x_{1})(x_{2} - x_{3})} f(x_{2}) + \frac{3 x^{2} - 2 x (x_{0} + x_{1} + x_{2}) + x_{0} (x_{1} + x_{2}) + x_{1} x_{2}}{(x_{3} - x_{0})(x_{3} - x_{1})(x_{3} - x_{2})} f(x_{3}) \end{equation*} $$

Segunda derivada $$ \begin{equation*} f''(x) = \frac{2[3 x - (x_{1} + x_{2} + x_{3})]}{(x_{0} - x_{1})(x_{0} - x_{2})(x_{0} - x_{3})} f(x_{0}) + \frac{2[3 x - (x_{0} + x_{2} + x_{3})]}{(x_{1} - x_{0})(x_{1} - x_{2})(x_{1} - x_{3})} f(x_{1}) + \frac{2[3 x - (x_{0} + x_{1} + x_{3})]}{(x_{2} - x_{0})(x_{2} - x_{1})(x_{2} - x_{3})} f(x_{2}) + \frac{2[3 x - (x_{0} + x_{1} + x_{2})]}{(x_{3} - x_{0})(x_{3} - x_{1})(x_{3} - x_{2})} f(x_{3}) \end{equation*} $$

Tercera derivada $$ \begin{equation*} f'''(x) = \frac{6}{(x_{0} - x_{1})(x_{0} - x_{2})(x_{0} - x_{3})} f(x_{0}) + \frac{6}{(x_{1} - x_{0})(x_{1} - x_{2})(x_{1} - x_{3})} f(x_{1}) + \frac{6}{(x_{2} - x_{0})(x_{2} - x_{1})(x_{2} - x_{3})} f(x_{2}) + \frac{6}{(x_{3} - x_{0})(x_{3} - x_{1})(x_{3} - x_{2})} f(x_{3}) \end{equation*} $$

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