Maths Revisions

Function, Calculus

Exponential Rules

\[a^m \times a^n = a^{m+n} ;\;\;\;\;\;\; a^{-m} = \frac{1}{a^m}\\ \frac{a^m}{a^n} = a^{m-n} ;\;\;\;\;\;\; (a^m)^n = a^{mn} \\ (ab)^m=a^mb^m ;\;\;\;\;\;\; a^0=1\]

Rules for Logarithms

\[ log(ab)=log(a)+log(b) ;\;\;\;\;\;\; log(\frac{a}{b})= log(a)-log(b) \\ log(a^n)=n\times log(a) ;\;\;\;\;\;\; log_aa=1 \\ log_a1=0 \]

Rules for the summation and product fuctions

\[\prod_{i=1}^n x_i=(x_1\cdot x_2\cdot x_3 \dots x_n) \\ \prod_{i=1}^nax_i = a^n\prod_{i=1}^nx_i \\ \sum_{i=1}^nx_i=(x_1+x_2+x_3+\dots+x_n) \\ \sum_{i=1}^nax_i=a\sum_{i=1}^nx_i \\ \sum_{i=1}^na=na \]

Some Rules for Defferentiation

\[\frac{d}{dx}a=0 ;\;\;\;\;\;\; \frac{d}{dx}ax=a \\ \frac{d}{dx}x^n=nx^{n-1} ;\;\;\;\;\;\; \frac{d}{dx}log_e(x)=\frac{1}{x}\\ \frac{d}{dx}e^x=e^x ;\;\;\;\;\;\; \frac{d}{dx}e^{F(x)}=\frac{dF(x)}{dx}e^{F(x)}=F^\prime(x)e^{F(x)}\\\frac{d}{dx}(F(x)+L(x))=F'(x)+L'(x)\\ \frac{d}{dx}(F(x)\cdot L(x))=L(x)F'(x)+F(x)L'(x)\\ \frac{d}{dx}(\frac{F(x)}{L(x)})=\frac{F'(x)}{L(x)}-\frac{F(x)\cdot L'(x)}{[L(x)]^2}=\frac{F'(x)L(x)-L'(x)F(x)}{[L(x)]^2}\]

Chain Rule: \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)

Some Rules for Integrals

\[\int a dx=ax ;\;\;\;\;\;\; \int af(x)dx = a \int f(x)dx \\ \int x^ndx = \frac{x^{n+1}}{n+1} ;\;\;\;\;\;\; \int F'(x)[F(x)]^n dx = \frac{[F(x)]^{n+1}}{n+1} \\ \int e^xdx = e^x ;\;\;\;\;\;\; \int F'(x)e^{F(x)}dx = e^{F(x)} \\ \int \frac{1}{x}dx = log_e(x) ;\;\;\;\;\;\; \int \frac{F'(x)}{F(x)}dx=log_e(F(x))\\ \int (F(x)+L(x))dx = \int F(x)dx + \int L(x)dx\]

Intergrating by parts: \(\int_a^b u\frac{dv}{dx}dx = [uv]_a^b - \int_a^bv\frac{du}{dx}dx\)

RULE: L. I. A. T. E. (分步積分法應用時定義 u 的優先順序)

  • L: logarithm
  • I: Inverse trigonometric (反三角函數)
  • A: Algebra
  • T: Trigonometric
  • E: Exponential

Related

comments powered by Disqus