Calculus/Answers to introductory concepts

Differentiation

Proof of the Derivative of a Constant Function

$\text{If }c=f(x)\text{, then}$
$\frac{d}{dx}\left[c\right]$	$=f^{\prime }(x)$
	$=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$
	$=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{c-c}{\mathrm{\Delta }x}$
	$=\underset{\mathrm{\Delta }x\to 0}{\lim }0$
	$=0$

Proof of the Derivative of a Linear Function

$\text{If }mx+b=f(x)\text{, then}$
$\frac{d}{dx}(mx+b)$	$=f^{\prime }(x)$
	$=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$
	$=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{[m(x+\mathrm{\Delta }x)+b]-[mx+b]}{\mathrm{\Delta }x}$
	$=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{mx+m\mathrm{\Delta }x+b-mx-b}{\mathrm{\Delta }x}$
	$=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{m\mathrm{\Delta }x}{\mathrm{\Delta }x}$
	$=\underset{\mathrm{\Delta }x\to 0}{\lim }m$
	$=m$

Proof of the Constant Multiple Rule

$\frac{d}{dx}[cf(x)]$	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{cf(x+\mathrm{\Delta }x)-cf\left(x\right)}{\mathrm{\Delta }x}$
	= $c\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$
	= $c\frac{d}{dx}[f(x)]$

Proof of the Addition and Subtraction Rules

$\frac{d}{dx}[f(x)\pm g(x)]$	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{[f(x+\mathrm{\Delta }x)+g(x+\mathrm{\Delta }x)]-[f(x)+g(x)]}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)+g(x+\mathrm{\Delta }x)-f(x)-g(x)}{\mathrm{\Delta }x}$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }[\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}+\frac{g(x+\mathrm{\Delta }x)-g(x)}{\mathrm{\Delta }x}]$
	= $\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}+\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{g(x+\mathrm{\Delta }x)-g(x)}{\mathrm{\Delta }x}$
	= $\frac{d}{dx}[f(x)]+\frac{d}{dx}[g(x)]$