SciTech-Mathmatics-Real Space + Taylor Equation + Exponential Functions+Trigonometrical Functions + Complex Space + Euler's Equation

Derivative and Slope

Quick review: a \(derivative\) gives us the \(\text{slope of a function}\) at \(any\ point\).

These derivative rules can help us:

• The derivative of \(a\ constant\) is 0
• The derivative of \(a x\) is \(a\) (example: the derivative of \(2x\) is \(2\))
• The derivative of \(x^n\) is \(nx^{n-1}\) (example: \(\text{the derivative of }x^3\text{ is }3x^2\))
• We will use the little mark \(’\) to denote "\(\text{derivative of}\)" (example: \(f'(x)\) denote the \(\text{derivative of } f(x)\)).

\(Real\ Space\) and \(Taylor\ Series\):

• \(\large \text{First define } f^{(0)} (x) = f(x) \text{ and } 0! = 1\) :
• Formula:
  
  \(\large \begin{array}{rll} \\
  f(x) &=& \sum_{n=0}^{\infty}{\frac{f^{(n)}(x_0)}{n!} (x-x_0)^n} \\
  &=& f(x_0) +\sum_{n=1}^{\infty}{\frac{f^{(n)}(x_0)}{n!} (x-x_0)^n} \\
  \end{array}\)
• Examples,\(Natural\ Exponential\ Functions\) and \(Trigonometrical\ Functions\):
  
  \(\large \begin{array}{rll} \\
  e^x &=& \sum_{n=0}^{\infty} {\frac{x^n}{n!}} \\
  &=& 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots + \frac{x^n}{n!} \\
  \cos (x) &=& \sum_{n=0}^{\infty} {\frac{(-1)^{(n)}}{(2n)!} {x^{(2n)}} } \\
  &=& 1 + \frac{(-1)x^{2}}{2!} + \frac{(+1)x^{4}}{4!} + \cdots + \frac{(-1)^{(n)}}{(2n)!} {x^{(2n)}} \\
  \sin (x) &=& \sum_{n=0}^{\infty} {\frac{(-1)^{n}}{(2n+1)!} {x^{(2n+1)}} } \\
  &=& x + \frac{(-1)x^{3}}{3!} + \frac{(+1)x^{5}}{5!} + \cdots + \frac{(-1)^{n}}{(2n+1)!} {x^{(2n+1)}} \\
  \end{array}\)

\(Complex\ Space\) and \(Euler's\ Equation\):

Let $\large \ x=i \cdot y $ and \(\large i^2=-1\):

\(\large \begin{array}{rll} \\
e^x = e^{i \cdot y} &=& \sum_{n=0}^{\infty} {\frac{(i \cdot y)^n}{n!}} \\
&=& 1 + i \cdot y + \frac{-1 \cdot y^2}{2!} + i \cdot \frac{-1 \cdot y^3}{3!} + \frac{+1 \cdot y^4}{4!} + i \cdot \frac{+1 \cdot y^5}{5!} + \cdots + i^{n} \cdot \frac{(y^n}{n!} \\
&=& (1 + \frac{-1 \cdot y^2}{2!} + \frac{+1 \cdot y^4}{4!} + \cdots ) + i ( y + \frac{-1 \cdot y^3}{3!} + \frac{+1 \cdot y^5}{5!} + \cdots ) \\
&=& \cos y + i \sin y \\
\therefore e^{ix} &=& \cos x + i \sin x\ , \text{ Euler's Equation} \\
\end{array}\)

So $\large the\ Taylor's\ Equation,\ Euler's\ Equation \text{ are unified in }Complex\ Space \text{ with } the\ Trigonometry\ Functions, the\ Natural\ Number \text{ and } Exponential\ Functions $