Slides - School of Engineering - University of Warwick

3.4 Second Order Derivative: ∂2 f
∂x 2
Similar formulae can be derived for second order or higher
order derivatives.
( ) ∂f
f i+1 = f i + ∆x
∂x
( ) ∂f
f i−1 = f i − ∆x
∂x
i
+ 1 ( ∂ 2 f
2! ∆x2 ∂x
)i
2
+ 1 ( ∂ 2 f
2! ∆x2 ∂x
)i
2
After minor rearrangement, we get
i
+ 1 3! ∆x3 ( ∂ 3 f
∂x 3 )i
− 1 3! ∆x3 ( ∂ 3 f
∂x 3 )i
+ H,
+ H.
∂ 2 f
∂x 2 = f i+1 − 2f i + f i−1
∆x 2
+ 1 12 ∆x2∂4 f
∂x 4 + H. (3.30FP )
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