View source for Calculus/Answers to introductory concepts

==Differentiation==
===Proof of the Derivative of a Constant Function===
{|
|-
|colspan="2"| <math>\mbox{If } c = f(x) \mbox{, then}\,</math>
|-
| <math>\frac{d}{dx} \left[\,c\right]</math> || <math>= f'(x)\,</math>
|-
| &nbsp; || <math>= \lim_{\Delta x \to 0} { f(x+\Delta x)-f(x) \over \Delta x }</math>
|-
| &nbsp; || <math>= \lim_{\Delta x \to 0} { c-c \over \Delta x }</math>
|-
| &nbsp; || <math>= \lim_{\Delta x \to 0} \,0</math>
|-
| &nbsp; || <math>= 0\,</math>
|-
|}

===Proof of the Derivative of a Linear Function===
{|
|colspan="2"| <math>\mbox{If } mx+b = f(x) \mbox{, then}\,</math>
|-
| <math>\frac{d}{dx} (mx+b)</math> || <math>= f'(x)\,</math>
|-
|  || <math>= \lim_{\Delta x \to 0} { f(x+\Delta x)-f(x) \over \Delta x }</math>
|-
|  || <math>= \lim_{\Delta x \to 0} { [m(x+\Delta x)+b]-[mx+b] \over \Delta x }</math>
|-
|  || <math>= \lim_{\Delta x \to 0} { mx+m\Delta x+b-mx-b \over \Delta x }</math>
|-
|  || <math>= \lim_{\Delta x \to 0} { m\Delta x \over \Delta x }</math>
|-
|  || <math>= \lim_{\Delta x \to 0} \,m</math>
|-
|  || <math>= m\,</math>
|-
|}

===Proof of the Constant Multiple Rule===
{|
|-
|<math>\frac{d}{dx}\left[cf(x)\right]</math>
| = <math>\lim_{\Delta x \to 0}\frac{cf\left(x+\Delta x \right)-cf\left(x\right)}{\Delta x}</math>
|-
|
| = <math>c\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}</math>
|-
|
| = <math>c\frac{d}{dx}\left[f(x)\right]\,</math>
|}

===Proof of the Addition and Subtraction Rules===
{|
|-
|<math>\frac{d}{dx}\left[f(x)\pm g(x)\right]</math>
| = <math>\lim_{\Delta x\to0}\frac{\left[f(x+\Delta x)+g(x+\Delta x)\right]-\left[f(x)+g(x)\right]}{\Delta x}</math>
|-
|
| = <math>\lim_{\Delta x\to 0}\frac{f(x+\Delta x)+g(x+\Delta x)-f(x)-g(x)}{\Delta x}</math>
|-
|
| = <math>\lim_{\Delta x\to 0}\left[\frac{f(x+\Delta x)-f(x)}{\Delta x}+\frac{g(x+\Delta x)-g(x)}{\Delta x}\right]</math>
|-
|
| = <math>\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}+\lim_{\Delta x\to 0}\frac{g(x+\Delta x)-g(x)}{\Delta x}</math>
|-
|
| = <math>\frac{d}{dx}\left[f(x)\right]+\frac{d}{dx}\left[g(x)\right]</math>
|-
|}

[[Category:Calculus]]