Derivatives can be approximated by finite differences in several ways, for instance

Assuming to be differentiable, in the limit , all three expressions are equivalent and exact, but for finite there are differences. Further discrepancies also arise when using finite precision arithmetic.

4.1 Estimate numerically the derivative of at using the three expressions given above and different step sizes . Use at least 50 logarithmically spaced values .

4.2 Display the absolute difference between the numerical results and the exact value of the derivative, against in a doubly logarithmic plot. You should format the plot so that the data presentation is clear and easy to understand.

4.3 Describe and interpret your results in no more than 250 words.

Hint: run the code below.

In [ ]:

h = 1e-14 print(1 + h - 1) print((1 + h - 1)/h)