# Derivatives Using Newton's Backward Difference Formula

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At x = x_{n-1}, we get s = – 1. Hence, we obtain the approximation to the second derivative f ″(x_{n-1}) as

...........................1.8

We use the formulas (1.4), (1.5), (1.7) and (1.8) when the entire data is to be used.

**Note : **We use the backward difference formulas for derivatives, when we need the values of the derivatives near the end of table of values.

**Example : **Using the operator relation, derive approximations to the derivatives f′(xn), f″(xn) in terms of the backward differences.

**Solution: **From the operator relation E = e^{hd}, where D = d/dx, we obtain

or

or