User talk:90.175.163.116

lagrange polynomial derivatives
Hi 90.175.163.116, and welcome to Wikipedia.

I added back the previous expressions to because they seem a bit less of a leap for readers than the simplified versions you added. If we are starting from:


 * $$\begin{aligned}

\ell_j(x) &= \prod_{\begin{smallmatrix}m = 0\\ m\neq j\end{smallmatrix}}^k \frac{x-x_m}{x_j-x_m}. \end{aligned}$$

Then the derivative is:


 * $$\begin{aligned}

\ell_j'(x) &= \sum_{\begin{smallmatrix}i=0 \\ i\not=j\end{smallmatrix}}^k \Biggl[ \frac{1}{x_j-x_i}\prod_{\begin{smallmatrix}m=0 \\ m\not = (i, j)\end{smallmatrix}}^k \frac{x-x_m}{x_j-x_m} \Biggr] \end{aligned}$$

As a consequence of the product rule.

We might think of the bit I added back as steps along the way to demonstrating your simpler expressions. Also, I think your second derivative expression is going to be less obvious to readers how to generalize to the 3rd, 4th, etc. derivatives compared to the first and second lines there.

Do you know a good source describing/deriving derivatives of lagrange polynomials? It would be even better for readers if we had a clear source to point at. –jacobolus (t) 19:51, 15 February 2023 (UTC)