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In analogy to the basic calculations involving derivatives, the following rules can be stated:

## Proposition: Basic Calculations Involving the Difference Operator

Let $D\subseteq\mathbb R$ ($D$ being a subset of real numbers). Let $x, x+1\in D,\lambda\in\mathbb R,$ and let $f,g:D\to\mathbb R.$ Then

1. $\Delta(f\pm g)(x)=\Delta f(x)\pm \Delta g(x),$
2. $\Delta (\lambda f)(x)=\lambda \Delta f(x),$
3. $\Delta (fg)(x)=g(x)\Delta f(x) + f(x+1)\Delta g(x)$ (also known as the product rule).
4. If $g(x+1)g(x)\neq 0$ for all $$x\in D$$, then (also known as the the quotient rule):
$$\Delta\left(\frac fg\right)(x)=\frac{g(x)\Delta f(x) – f(x)\Delta g(x)}{g(x+1)g(x)}.$$

| | | | | created: 2020-03-23 21:20:40 | modified: 2020-03-24 09:20:06 | by: bookofproofs | references: [1112], [8404], [8405]