Step 1 - Data and Initialization:

You have a function f(x) for which you want to calculate the derivative at a specific point x.

Step 2 - Choose a Difference Formula:

Select a difference formula to approximate the derivative. Common choices include the forward difference, backward difference, and central difference formulas.

Step 3 - Apply the Chosen Difference Formula:

Use the selected difference formula to estimate the derivative. The formulas are as follows:

Forward Difference:

f'(x) ≈ (f(x + h) - f(x)) / h

Backward Difference:

f'(x) ≈ (f(x) - f(x - h)) / h

Central Difference:

f'(x) ≈ (f(x + h) - f(x - h)) / (2h)

Here, h is a small step size or interval. The smaller h is, the more accurate the approximation, but it can introduce numerical instability if too small.

Step 4 - Determine the Accuracy and Convergence:

You can assess the accuracy of the numerical differentiation by calculating the error, typically using higher-order terms in the Taylor series expansion.

Step 5 - Output:

The result of the chosen difference formula is an approximation of the derivative of f(x) at the point x.

Numerical differentiation is used when the analytical form of a function's derivative is not available or is difficult to compute. It provides a straightforward way to estimate derivatives at specific points.