<jsm>\frac{1}{h}(u(x+h)-u(x))=u'(x)+\frac{h}{2}u(x)+\frac{h^2}{3!}u'(x)+\mathcal{O}(h^3).</jsm>

<jsm>\frac{1}{h}(u(x)-u(x-h))=u'(x)-\frac{h}{2}u(x)+\frac{h^2}{3!}u'(x)+\mathcal{O}(h^3).</jsm>

<jsm>\frac{1}{2h}(u(x+h)-u(x-h))=u'(x)+\frac{h^2}{3!}u'''(x)+\mathcal{O}(h^3).</jsm>

<jsm>\frac{1}{h^2}\delta^2\,u(x)=\frac{1}{h^2}(u(x+h)-2u(x)+u(x-h)),</jsm> og ved Taylor utvikling

<jsm>\frac{1}{h^2}\delta^2\,u(x)=\frac{1}{h^2}[u(x)+hu'(x)+\frac{h^2}{2}u^{(2)}(x)+\frac{h^3}{3!}u^{(3)}(x)+ \frac{h^4}{4!}u^{(4)}(x)</jsm>

<jsm>\hskip2.7cm-2u(x)+u(x)-hu'(x)+\frac{h^2}{2}u^{(2)}(x)-\frac{h^3}{3!}u^{(3)}(x)+ \frac{h^4}{4!}u^{(4)}(x)+\mathcal{O}(h^5)]</jsm>

<jsm>\frac{1}{h^2}\delta^2\,u(x)=u^{(2)}(x)+\frac{h^2}{12}u^{(4)}(x)+\mathcal{O}(h^4).</jsm>

En tilnærmelse til den tredje deriverte er gitt av en kombinasjon av forover differanse og sentral diffeanse i andre:

<jsm>\frac{1}{h}\Delta\,\frac{1}{h^2}\delta^2\,u(x) </jsm>

gir

<jsm>\frac{1}{h}\Delta\,\frac{1}{h^2}\delta^2\,u(x)=\frac{1}{h^3}(u(x+2h)-2u(x+h)+u(x)-u(x+h)+2u(x)-u(x-h)</jsm>

og

<jsm>\frac{1}{h}\Delta\,\frac{1}{h^2}\delta^2\,u(x)=\frac{1}{h^3}(u(x+2h)-3u(x+h)+3u(x)-u(x-h)).</jsm>

For å finne hvordan diffeanseformelen approksimerer den tredje deriverte, bruker vi kunnskapet fra tilnærmelse av den andre deriverte (forrige oppgave), vi får:

<jsm>\frac{1}{h}\Delta\,\frac{1}{h^2}\delta^2\,u(x)=\frac{1}{h}\Delta (u^{(2)}(x)+\frac{1}{12}h^2 u^{(4)}(x)+\mathcal{O}(h^4)),</jsm>

<jsm>\frac{1}{h}\Delta\,\frac{1}{h^2}\delta^2\,u(x)=\frac{1}{h} (u^{(2)}(x+h)+\frac{1}{12}h^2 u^{(4)}(x+h)-u^{(2)}(x)-\frac{1}{12}h^2 u^{(4)}(x)+\mathcal{O}(h^4)), </jsm>

og ved bruk av Taylor utvikling

<jsm>\frac{1}{h}\Delta\,\frac{1}{h^2}\delta^2\,u(x)=\frac{1}{h} (u^{(2)}(x)+hu^{(3)}(x)+\frac{h^2}{2}u^{(4)}(x)+\frac{1}{12}h^2u^{(4)}(x)-u^{(2)}(x)-\frac{1}{12}h^2u^{(4)}(x)+\mathcal{O}(h^3)),</jsm>

<jsm>\frac{1}{h}\Delta\,\frac{1}{h^2}\delta^2\,u(x)=u^{(3)}(x)+\frac{h}{2}u^{(4)}(x)+\mathcal{O}(h^2).</jsm>