Discrete Calculus

Discrete calculus, sometimes called Finite calculus, is the study of incremental change, as opposed to regular calculus, the study of continuous change. Discrete calculus most regularly deals with solving summations and functions defined on the natural numbers.

Discrete Derivative

In continuous calculus, the derivative is the ‘instantaneous rate of change’ of a function. That concept can’t exist with incremental change. The smallest rate of change in a discrete function is the change between integers. Therefore we define our discrete derivative

Definition

The discrete derivative is the smallest change in a discrete function, defined $$ f(x) = f(x+1) - f(x)

We can now prove may properties about the discrete derivative

Theorem

The discrete derivative has the following properties

Constant rule

\Delta c = 0 for all c \in \mathbb{N}

Linearity

\Delta af(x) = a \Delta f(x)

Product rule

\Delta f(x)g(x) = f(x) \Delta g(x) + g(x) \Delta f(x) + \Delta f(x) \Delta g(x)

Proof

To-do

Functions

Now that we’ve defined the discrete derivative, we can start finding the discrete derivative of functions.

Take for example, the function