# Numerical Differentiation

Numerical differentiation is differentiating a function in a way that does not involve normal differentiating methods. This may be because the function is too complex to differentiate directly.

## Finite Difference Approximations

This type of differentiation involves taking the point at which you want to differentiate, and find the slope using either a point before, after or both. Note that this only finds an approximation of the derivative, and at only one point at a time.

The following are the equations to calculate the derivative. These are to calculate the derivative at x, where f_{i+1} = f(x+h) and f_{i-1} = f(x-h), given that h = Δx. Note, a smaller Δx will give a more accurate solution.

Second derivative using Central Difference:

#### Errors

The error for forward and backward difference is O(h), meaning a first degree truncation error. The higher the degree, the more accurate the approximation. Central difference is O(h^{2}), and is therefore more accurate. This is because it takes the average of forward and backward difference.

## End

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