<Dl> <Dd> R (x) ≤ (x n − x 0) n + 1 (n + 1)! max x 0 ≤ ξ ≤ x n f (n + 1) (ξ). (\ displaystyle R (x) \ leq (\ frac ((x_ (n) - x_ (0)) ^ (n + 1)) ((n + 1)!)) \ max _ (x_ (0) \ leq \ xi \ leq x_ (n)) f ^ ((n + 1)) (\ xi).) </Dd> </Dl> <Dd> R (x) ≤ (x n − x 0) n + 1 (n + 1)! max x 0 ≤ ξ ≤ x n f (n + 1) (ξ). (\ displaystyle R (x) \ leq (\ frac ((x_ (n) - x_ (0)) ^ (n + 1)) ((n + 1)!)) \ max _ (x_ (0) \ leq \ xi \ leq x_ (n)) f ^ ((n + 1)) (\ xi).) </Dd> <P> The first derivative of the lagrange polynomial is given by </P> <Dl> <Dd> L ′ (x): = ∑ j = 0 k y j l j ′ (x) (\ displaystyle L' (x): = \ sum _ (j = 0) ^ (k) y_ (j) \ ell _ (j)' (x)) </Dd> </Dl>

State and prove lagrange's formula for interpolation