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Programmatic Numerical Approximation of a Derivative
To implement the numerical approximation of a derivative in code, we evaluate the finite difference ratio for progressively smaller values of . For example, consider the function . We can define this function in Python and calculate its numerical derivative at as the perturbation shrinks exponentially:
import numpy as np def f(x): return 3 * x ** 2 - 4 * x for h in 10.0**np.arange(-1, -6, -1): print(f'h={h:.5f}, numerical limit={(f(1+h)-f(1))/h:.5f}')
As approaches , the printed output shows the numerical limit correctly converging to the exact analytical derivative value, .
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Programmatic Numerical Approximation of a Derivative