Basics of Maths

less than 1 minute read

Linear Algebra

Calculus

  • Differentiation rules

    • Addition/subtraction

      \[\begin{aligned} f(x) &= g(x) \pm h(x) \\ f'(x) = \dfrac{df}{dx} &= g'(x) \pm h'(x) \end{aligned}\]
    • Multiplication

      \[\begin{aligned} f(x) &= g(x) * h(x) \\ f'(x) = \dfrac{df}{dx} &= g(x)*h'(x) + h(x)*g'(x) \end{aligned}\]
    • Division

      \[\begin{aligned} f(x) &= \dfrac{g(x)}{h(x)} \\ f'(x) = \dfrac{df}{dx} &= \dfrac{h(x)g'(x)-g(x)h'(x)}{h(x)^2} \end{aligned}\]
  • Taylor Series

    • Taylor series can approximate any function $f(x)$ around some $x$

      \[\begin{aligned} f(x+h) &= \sum_{i=0}^{\infin} \frac{h^i}{i!}f^{i}(x) \qquad \forall x \in \textbf{dom } f \\ f(x+h) &= f(x) + hf'(x) + \frac{h^2}{2!}f''(x) + \dots \end{aligned}\]
  • L’Hopital rule

    • if $ \lim_{x \to a} \dfrac{f(x)}{g(x)} $ is of the form $\dfrac{0}{0}$ or $\dfrac{\infin}{\infty}$, then,

      \[\lim_{x \to a} \dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f'(x)}{g'(x)} \quad \forall a \in \R\]
  • Basic

    \[\nabla f(x) \rightarrow grad of f(x)\]

References

  1. https://www.geogebra.org/graphing

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