בדומה לפיתוחים הנ"ל, נמצא נוסחאות מקורבות לנגזרות:
קירוב לינארי:
קירוב ריבועי:
h 2 f ″ ( x i ) = [ − ln ( 1 − ∇ ) ] 2 = [ ∇ 2 + ∇ 3 + 11 12 ∇ 4 + 5 6 ∇ 5 + 137 180 ∇ 6 + . . . ] f ( x i ) {\displaystyle \ h^{2}f''(x_{i})\ =\ \left[-\ln(1-\nabla )\right]^{2}\ =\ \left[\nabla ^{2}+\nabla ^{3}+{11 \over 12}\nabla ^{4}+{5 \over 6}\nabla ^{5}+{137 \over 180}\nabla ^{6}+...\right]f(x_{i})}
![{\displaystyle \ =-\ln(1-\nabla )f(x_{i})=\left[\nabla +{\nabla ^{2} \over 2}+{\nabla ^{3} \over 3}+{\nabla ^{4} \over 4}+...\right]f(x_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba311f0cb9819f6cc838233a816085464f505711)
![{\displaystyle \ f'(x_{i})\approx {1 \over h}\nabla f(x_{i})={1 \over h}[f(x_{i})-f(x_{i-1})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/967b974cc2fc9598b69bb6e3eec060436eff6375)
![{\displaystyle \ =\left[\nabla +{\frac {\nabla ^{2}}{2}}+{\frac {2\nabla ^{3}}{3!}}+...\right]\left[1+\theta \nabla +{\frac {\theta (\theta +1)}{2}}\nabla ^{2}+...\right]f(x_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71f7298d97fa723283d1817c78a7491ba97e1ebc)
![{\displaystyle \ \Rightarrow \quad f'(x_{0}+\theta h)={1 \over h}\left[\nabla +{\frac {2\theta +1}{2}}\nabla ^{2}+{\frac {3\theta ^{2}+6\theta +2}{6}}\nabla ^{3}+...\right]f(x_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88226e2c04e0be6522147984ce3fb6c26a663bb8)
![{\displaystyle \ h^{2}f''(x_{i})\ =\ \left[-\ln(1-\nabla )\right]^{2}\ =\ \left[\nabla ^{2}+\nabla ^{3}+{11 \over 12}\nabla ^{4}+{5 \over 6}\nabla ^{5}+{137 \over 180}\nabla ^{6}+...\right]f(x_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5829ac03b53e7faea2420537f1117724bf1f9db8)
![{\displaystyle \ f''(x_{i})={1 \over h^{2}}[f(x_{i-2})-2f(x_{i-1})+f(x_{i})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25484d0909dbb66941f1f623f60e9510fd9e913e)
![{\displaystyle \ f'(x_{0}+\theta h)={1 \over h^{2}}\left[\nabla +{\frac {\nabla ^{2}}{2}}+{\frac {2\nabla ^{3}}{3!}}+...\right]\left[\nabla +{\frac {2\theta +1}{2}}\nabla ^{2}+{\frac {3\theta ^{2}+6\theta +2}{6}}\nabla ^{3}+...\right]f(x_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72a995eec2bdaaf1e75b20043023fc0380dceb2a)
![{\displaystyle \ \Rightarrow \quad f''(x_{0}+\theta h)={1 \over h^{2}}\left[\nabla ^{2}+(\theta +1)\nabla ^{3}+{\frac {6\theta ^{2}+18\theta +11}{12}}\nabla ^{4}+...\right]f(x_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31f55b5f4da5da7f900e3bd3dcd65731c6816fc5)