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

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

- $\Delta(f\pm g)(x)=\Delta f(x)\pm \Delta g(x),$
- $\Delta (\lambda f)(x)=\lambda \Delta f(x),$
- $\Delta (fg)(x)=g(x)\Delta f(x) + f(x+1)\Delta g(x)$ (also known as the
**product rule**). - 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]

[8405] **Bool, George**: “A Treatise on the Calculus of Finite Differences”, Dover Publications, Inc., 0

[1112] **Graham L. Ronald, Knuth E. Donald, Patashnik Oren**: “Concrete Mathematics”, Addison-Wesley, 1994, 2nd Edition

[8404] **Miller, Kenneth S.**: “An Introduction to the Calculus of Finite Differences And Difference Equations”, Dover Publications, Inc, 0