differentiation_of_continuous_function.pdf

As shown above both forward and backward divided

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As shown above, both forward and backward divided difference approximation of the first derivative are accurate on the order of x 0 Can we get better approximations? Yes, another method to approximate the first derivative is called the Central difference approximation of the first derivative .
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19 Derive the forward difference approximation from Taylor series Cont. From Taylor series 3 2 1 Δ ! 3 Δ ! 2 Δ x x f x x f x x f x f x f i i i i i 3 2 1 Δ ! 3 Δ ! 2 Δ x x f x x f x x f x f x f i i i i i Subtracting equation (2) from equation (1)  3 1 1 Δ ! 3 2 Δ 2 x x f x x f x f x f i i i i 2 1 1 ! 3 2 x x f x x f x f x f i i i i 2 1 1 0 2 x x x f x f x f i i i
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20 Central Divided Difference Hence showing that we have obtained a more accurate formula as the error is of the order of . 2 Δ 0 x x f(x) x- Δx x x+Δx Figure 3 Graphical Representation of central difference approximation of first derivative
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