Numerical Differentiation

Updated 02 Oct 2026

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What it is
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  approximating a function's derivative using discrete function evaluations rather than symbolic differentiation
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Forward difference
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  $f'(x) \approx \dfrac{f(x+h)-f(x)}{h}$ ; truncation error is order $O(h)$
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  total error bound (truncation + rounding): $E_{total}(h) \leq \dfrac{Mh}{2} + \dfrac{2\epsilon}{h}$ , where $M$ bounds $|f''(x)|$ and $\epsilon$ bounds the function's rounding error
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Central difference
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  $f'(x) \approx \dfrac{f(x+h)-f(x-h)}{2h}$ ; truncation error is order $O(h^2)$ - subtracting $f(x-h)$ from $f(x+h)$ cancels the even-power Taylor terms. After dividing by $2h$ , the leading truncation error is $O(h^2)$ .
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  total error: $E_{central}(h) \approx \dfrac{M_3 h^2}{6} + \dfrac{\epsilon}{h}$ , where $M_3$ bounds $|f'''(x)|$
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Second derivative (central difference)
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  $f''(t) \approx \dfrac{f(t+h)-2f(t)+f(t-h)}{h^2}$ ; dividing by $h^2$ amplifies input noise/uncertainty significantly (e.g. by 100× for $h=0.1$ ) — a caution when differentiating noisy sensor data
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Related
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  Optimal Step Size
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  Richardson Extrapolation

Linked references 5