As shown above, both forward and backward divided differenceapproximation of the first derivative are accurate on the order of x0Can we get better approximations? Yes, another method to approximate the first derivative is called the Central difference approximation ofthe first derivative.
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19Derive the forward difference approximation from Taylor series Cont.From Taylor series321Δ!3Δ!2Δxxfxxfxxfxfxfiiiii321Δ!3Δ!2ΔxxfxxfxxfxfxfiiiiiSubtracting equation (2) from equation (1)311Δ!32Δ2xxfxxfxfxfiiii211!32xxfxxfxfxfiiii21102xxxfxfxfiii
20Central Divided DifferenceHence showing that we have obtained a more accurate formula as the error is of the order of .2Δ0xxf(x)x-Δx x x+ΔxFigure 3 Graphical Representation of central difference approximation of first derivative