Mathematical Identities

Mathematical Identities

Wikipedia Article[1]

Perfect Square: A number, whose square root is an integer. Example: 1, 4, 16, 25.

Substitution (Quantitative Comparisons): When substituting in quantitative comparison problems, don’t rely on only positive whole numbers. You must also check negative numbers, fractions, 0, and 1 because they often give results different from those of positive whole numbers. Plug in the numbers 0, 1, 2, –2, and 1/2, in that order.

Derivatives

Derivative Identities $$\frac{dy}{dx}=y^{'}$$ $$\frac{dy}{dx}=\frac{d}{dx}(y)$$ \(y^{'}=\frac{dy}{dx}=D_{x}\)

Radical Derivatives $$\sqrt{x}=x^{\frac{1}{2}}=\frac{1}{2\sqrt{x}}\:dx$$ $$\sqrt[3]{x^{2}} =x^{\frac{2}{3}}=\frac{2}{3\sqrt[3]{x}}\:dx$$ $$\sqrt[4]{x^{5}} =x^{\frac{5}{4}}=\frac{5}{4}\sqrt[4]{x}\:dx$$ $$\frac{1}{x^{\frac{1}{3}}} =x^{\frac{-1}{3}}=-\frac{1}{3x^{\frac{4}{3}}}\:dx$$ $$\frac{1}{x^{\frac{2}{3}}} =x^{\frac{-3}{2}}=-\frac{3}{2x^{\frac{5}{2}}}\:dx$$

Differentiation Rules
Constants and Single Variables
Rule	Name	Rule	Name	Rule	Name
$$\frac{d}{dx}c=0$$	Constnat	$$\frac{d}{dx}x=1$$		$$\frac{d}{dx}cx=c$$
$$\frac{d}{dx}x^{n}=nx^{n-1}$$		$$\frac{d}{dx}c^{x}=(\ln c)c^{x}$$		$$\frac{d}{dx}(\ln x)=\frac{1}{x}$$
$$\frac{d}{dx}e^{x}=e^{x}$$
Functions
Rule	Name	Rule	Name	Rule	Name
$$\frac{d}{dx}cu=cu^{'}$$		$$\frac{d}{dx}(u\pm v)=u^{'}\pm v^{'}$$	Distributive	$$\frac{d}{dx}uv=uv^{'}+vu^{'}$$	Product
$$\frac{d}{dx}\frac{u}{v}=\frac{vu^{'}-uv^{'}}{v^{2}}$$	Quotient	$$\frac{d}{dx}u^{n}=nu^{n-1}u^{'}$$		$$\frac{d}{dx}\left | u \right |=\frac{u}{\left | u \right |}u^{'},u\neq 0$$
$$\frac{d}{dx}\ln u=\frac{u^{'}}{u}$$		$$\frac{d}{dx}e^{u}=e^{u}u^{'}$$		$$\frac{d}{dx}\log_{c}u=\frac{u^{'}}{(\ln c)u}$$
$$\frac{d}{dx}c^{u}=(\ln\:c)c^{u}u^{'}$$		$$\frac{d}{dx}f(g(x))=f^{'}(g(x))g^{'}(x)$$	Chain Rule
Trigonometry
Rule	Name	Rule	Name	Rule	Name
$$\frac{d}{dx}\sin u=(\cos u)u^{'}$$		$$\frac{d}{dx}\cos u=-(\sin u)u^{'}$$		$$\frac{d}{dx}\tan u=(\sec^{2}u)u^{'}$$
$$\frac{d}{dx}\cot u=-(\csc^{2}u)u^{'}$$		$$\frac{d}{dx}\sec u=-(\sec u \tan u)u^{'}$$		$$\frac{d}{dx}\csc u=-(\csc u \cot u)u^{'}$$
$$\frac{d}{dx}\arcsin u=\frac{u^{'}}{\sqrt{1-u^{2}}}$$		$$\frac{d}{dx}\arccos=\frac{-u^{'}}{\sqrt{1-u^{2}}}$$		$$\frac{d}{dx}\arctan u =\frac{u^{'}}{1+u^{2}}$$
$$\frac{d}{dx} \text{arccot}\:u=\frac{-u^{'}}{1+u^{2}}$$		$$\frac{d}{dx}\text{arcsec}\:u=\frac{u^{'}}{\left | u \right |\sqrt{u^{2}-1}}$$		$$\frac{d}{dx}\text{arccsc} \:u=\frac{-u^{'}}{\left | u \right |\sqrt{u^{2}-1}}$$
$$\frac{d}{dx}\sinh u=(\cosh u)u^{'}$$		$$\frac{d}{dx}\cosh u=(\sinh u)u^{'}$$		$$\frac{d}{dx}\tanh u=(\text{sech}^{2}\: u)u^{'}$$
$$\frac{d}{dx}\text{coth}\: u=-(\text{csch}^{2}\: u)u^{'}$$		$$\frac{d}{dx}\text{sech}\: u=-(\text{sech}\:u \tanh u) u^{'}$$		$$\frac{d}{dx}\text{csch}\: u=-(\text{csch}\: u\: \text{coth}\: u)u^{'}$$
$$\frac{d}{dx}\sinh^{-1} u=\frac{u^{'}}{\sqrt{u^{2}+1}}$$		$$\frac{d}{dx}\cosh^{-1} u=\frac{u^{'}}{\sqrt{u^{2}-1}}$$		$$\frac{d}{dx}\tanh^{-1} u=\frac{u^{'}}{1-u^{2}}$$
$$\frac{d}{dx}\text{coth}^{-1}\: u=\frac{u^{'}}{1-u^{2}}$$		$$\frac{d}{dx}\text{sech}^{-1}\: u=\frac{-u^{'}}{u\sqrt{1-u^{2}}}$$		$$\frac{d}{dx}\text{csch}^{-1}\: u=\frac{-u^{'}}{\left | u \right |\sqrt{1+u^{2}}}$$

Integration

Integration Rules
Constants and Single Variables
Rule	Name	Rule	Name
$$\int dx=x+C$$		$$\int \frac{1}{x}dx=\ln \left | x \right |+C$$
$$\int x^{n}\:dx=\frac{x^{n+1}}{n+1}+C, n \ne -1$$		$$\int e^{x}dx=e^{x}+C$$
$$\int c^{x}dx=(\frac{1}{\ln c})c^{x}+C$$		$$\int x\:dy=xy - \int y\:dx$$
$$\int \ln x \:dx= x\: \ln x - x +C$$
Functions
Rule	Name	Rule	Name
$$\int c\:f(x)dx=c\:\int f(x)dx$$		$$\int \left [ f(x) \pm g(x) \right ]dx=\int f(x)dx \pm \int g(x) dx$$
$$(1+e^{x})^{2}=1+2e^{x}+e^{2x}$$	Expand Numerator	$$\frac{1+x}{x^{2}+1}=\frac{1}{x^{2}+1}+\frac{x}{x^{2}+1}$$	Separate Numerator
$$\frac{1}{\sqrt{2x-x^{2}}}=\frac{1}{\sqrt{1-(x-1){2}}}$$	Complete the Square	$$ \frac{x^{2}}{x^{2}+1}=1-\frac{1}{x^{2}+1}$$	Divide improper rational function
$$\frac{2x}{x^{2}+2x+1}$$ $$=\frac{2x+2-2}{x^{2}+2x+1}$$ $$=\frac{2x+2}{x^{2}+2x+1}-\frac{2}{(x+1)^{2}} $$	Add and subtract terms in the numerator	$$\frac{1}{1+\sin x} $$ $$=\bigg( \frac{1}{1+\sin x} \bigg) \bigg( \frac{1-\sin x}{1-\sin x} \bigg)$$ $$=\frac{1-\sin x}{1-\sin^{2} x}$$ $$=\frac{1-\sin x}{\cos^{2} x}$$ $$=\sec^{2}x - \frac{\sin x}{\cos^{2} x}$$	Multiply and divide by the Pythagorean conjugate.
Trigonometry
Rule	Name	Rule	Name
$$\int \sin x\:dx = -\cos x + C$$		$$\int \cos x\:dx = \sin x + C$$
$$\int \tan x\:dx = \ln \left | \cos x \right | + C$$		$$\int \cot x\:dx = \ln \left | \sin x \right | + C$$
$$\int \sec x\:dx = \ln \left | \sec x + \tan x \right | + C$$		$$\int \csc x\:dx = -\ln \left | \csc x + \cot x \right | + C$$
$$\int \sec^{2}x\:dx = \tan x + C$$		$$\int \csc^{2}x\:dx = - \cot x + C$$
$$\int \sec x \tan x\:dx = \sec x +C$$		$$\int \csc x \: \cot x\:dx = - \csc x + C$$
$$\int \frac {dx}{\sqrt{y^{2} - x^{2}}} = \text{arcsin} \frac{x}{y} + C$$		$$\int \frac {dx}{y^{2} + x^{2}}= \frac{1}{y} \text{arctan} \frac{x}{y} + C$$
$$\int \frac{dx}{x\sqrt{x^{2} - y^{2}}} = \frac{1}{y} \text{arcsec} \frac{\left | x \right |}{y}+ C$$

Factors

Exponents and Logarithms

Fractions

Number Theory

For any number p.
p = qz +r, where z is the quotent, r the remainder.
A Prime Number is a positive integer evenly divisible only by itself and 1.
A number is divisible evenly by 3 if the sum of it's digits is divisible by 3.
A number raised to an even exponent is greater than or equal to zero.

Properties of Odd and Even Numbers

Odd/Even applies to positive or negative integers.
A number is even if the remainder is 0 when divided by 2.
A number is odd if the remainder > 0 when divided by 2.
Zero is even
even * even = even
odd * odd = odd
even * odd = even

even +- even = even
odd +- odd = even
even +- odd = odd

even a / even b = a < b even fraction a > b even integer
even a / odd b = a < b repeating decimal a > b repeating decimal
odd a / odd b = a < b repeating odd decimal a > b odd integer or repeating odd decimal
odd a / even b = a < b terminating odd decimal a > b terminating odd decimal

Divisibility

All even numbers are evenly divisible by 2. By definition.
All numbers >= 5 that end in 0 or 5 are evenly divisible by 5.

Checking divisibility by adding up digits.
the Digital Root for a number is the digits sum.
For Example: 122 = 1+2+2 = 5.
Repeat the process until there is one digit.
For Example: 87,428 = 8 + 7 + 4 + 2 + 8 = 29 2 + 9 = 11 1 + 1 = 2
Every number whose digital root is 3,6,9 is evenly divisible by 3.
Every number whose digital root is 9 is evenly divisible by 9.
Basic Math for Dummies p. 118

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Parent Article: Mathematics