In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function. Numerical differentiation is the process of finding the numerical value of a derivative of a given function at a given point. In general, numerical differentiation is more difficult than numerical integration. This is because while numerical integration requires only good continuity properties of the function being integrated, numerical differentiation requires more complicated properties such as Lipschitz classes. there are several reasons as of why we still need to approximate derivatives: • Even if there exists an underlying function that we need to differentiate, we might know its values only at a sampled data set without knowing the function itself. • There are some cases where it may not be obvious that an underlying function exists and all that we have is a discrete data set. We may still be interested in studying changes in the data, which are related, of course, to derivatives. • There are times in which exact formulas are available but they are very complicated to the point that an exact computation of the derivative requires a lot of function evaluations. It might be significantly simpler to approximate the derivative instead of computing its exact value. • When approximating solutions to ordinary (or partial) differential equations, we typically represent the solution as a discrete approximation that is defined on a grid. Since we then have to evaluate derivatives at the grid points, we need to be able to come up with methods for approximating the derivatives at these points, and again, this will typically be done using only values that are defined on a lattice. The underlying function itself (which in this cased is the solution of the equation) is unknown. A simple approximation of the first derivative is f 0 (x) ≈ f(x + h) − f(x) h, where we assume that h > 0. What do we mean when we say that the expression on the right-hand-side of is an approximation of the derivative? For linear functions is actually an exact expression for the derivative.