The backward difference formulas make use of the values of the function to the left of the
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The backward difference formulas make use of the values of the function to the left of the base grid point. Accordingly, the first derivative at point \(i\left(t=t_{i}\right)\) is defined as
\[\frac{d x}{d t}=\frac{x(t)-x(t-\Delta t)}{\Delta t}=\frac{x_{i}-x_{i-1}}{\Delta t}\]
Derive the backward difference formulas for \(\left(d^{2} x\right) /\left(d t^{2}\right),\left(d^{3} x\right) /\left(d t^{3}\right)\), and \(\left(d^{4} x\right) /\left(d t^{4}\right)\) at \(t_{i}\).
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