proof of properties of derivatives by pure algebra

Theorem 1.

The derivative satisfies the following rules:

• $$\frac{d}{dx}(f(x)+g(x))=\frac{df}{dx}+\frac{dg}{dx},\frac{d}{dx}(af(x))=a\frac{df}{dx},$$

  for $f(x),g(x)\in R[x]$ and $a\in R$.

• $$\frac{d}{dx}(x^n)=n{x}^{n-1}.$$

• $$\frac{d}{dx}(f(x)g(x))=\frac{df}{dx}g(x)+f(x)\frac{dg}{dx}.$$

Remark 2.

The following proofs apply to derivatives by pure algebra (http://planetmath.org/DerivativesByPureAlgebra). While the nature of the proofs are similar to the usual proofs, the notion of a limit is replaced by modular arithmetic in $R\mathit{}\mathrm{[}x\mathrm{,}h\mathrm{]}\mathrm{/}\mathrm{(}h\mathrm{)}$.

Proof.

Power rule.

${\displaystyle \frac{d}{dx}}(x^n)$ $\equiv$ ${\displaystyle \frac{{(x+h)}^n-x^n}{h}}$

$=$ ${\displaystyle \sum _{j=1}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{j}}\right)x^{n-j}{h}^{j-1}$

$\equiv$ $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)x^{n-1}=n{x}^{n-1}.$

Linearity rule. For all $f(x),g(x)\in R[x]\cong R[x,h]/(h)$, it follows

$$\frac{(f+g)(x+h)-(f+g)(x)}{h}\equiv \frac{f(x+h)+g(x+h)-f(x)-g(x)}{h}\equiv \frac{f(x+h)-f(x)}{h}+\frac{g(x+h)-g(x)}{h}.$$

Furthermore, for all $a\in R$

$$\frac{(af)(x+h)-(af)(x)}{h}\equiv \frac{af(x+h)-af(x)}{h}=a\frac{f(x+h)-f(x)}{h}.$$

Product rule. In $R[x,h]$ modulo $(h)$ we have:

${\displaystyle \frac{d}{dx}}(fg)$ $\equiv$ ${\displaystyle \frac{f(x+h)g(x+h))-f(x)g(x)}{h}}$

$\equiv$ ${\displaystyle \frac{f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}}$

$\equiv$ ${\displaystyle \frac{(f(x+h)-f(x))g(x+h)+f(x)(g(x+h)-g(x))}{h}}$

$\equiv$ ${\displaystyle \frac{f(x+h)-f(x)}{h}}g(x+h)+f(x){\displaystyle \frac{g(x+h)-g(x)}{h}}$

$\equiv$ ${\displaystyle \frac{df}{dx}}g(x)+f(x){\displaystyle \frac{dg}{dx}}.$

∎

Title	proof of properties of derivatives by pure algebra
Canonical name	ProofOfPropertiesOfDerivativesByPureAlgebra
Date of creation	2013-03-22 16:00:03
Last modified on	2013-03-22 16:00:03
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	5
Author	Algeboy (12884)
Entry type	Proof
Classification	msc 26B05
Classification	msc 46G05
Classification	msc 26A24
Related topic	RulesOfCalculusForDerivativeOfPolynomial