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The plot displays the absolute error when computing the derivative of ${\displaystyle f(x)=x^3}$ (${\displaystyle f^{\prime }(x)=3x^2}$) using a forward difference (of first order) and a centered difference (of second order),

${\displaystyle \frac{f(x+h)-f(x)}{h}-3x^2}$

for various values of ${\displaystyle x}$. The centered difference is for ${\displaystyle x=0.1}$ only.

The noisy negative slope is due to finite precision (64-bit IEEE754) in the computer used, and the positive slope is due to the formula error of the differentiation formula.

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