Updated in 2026 : May : 12 : Mathematical Calculations ;

Updated on 12 : 5 : 2026 : Mathematical Calculations ;

(equation, expression, form, formula, function, method, system) Radical992;

( relations, relations, relations), also see: Control Character; Display .robotic LCD; Internet Connection; Mathematical Calculations; Server;

(( Radian, Radian, Radian, Radian), ( RADIAN, RADIAN), radian, ( RADIAN, RADIAN), ( Radian, Radian, Radian, Radian)), also see: Eccentric _circles; Gene Therapy System; G D C; imaginary_ pairs_ of_ moon_ waves_ for_ our_ earth; Manmade Global Weather; Mathematical Calculations; Nanobot Programming; 6_Planets_Prediction; radian; r Character;  2rComputer; Schematic Light;

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Basic trigonometric functions (SOH CAH TOA), e.g.

	s					(sine,	e.g. sine degree angle is equal to (opposite/hypotenuse)	6sComputer
	S					SINE) =
	o					(opposite,
	O					OPPOSITE)
	h					(hypotenuse,
	H					HYPOTENUSE)
	c					(cosine,	e.g. cosine degree angle is equal to (adjacent/hypotenuse)
	C					COSINE) =
	a					(adjacent,
	A					ADJACENT)		3cComputer
	h					(hypotenuse,
	H					HYPOTENUSE)
	t					(tangent,	e.g. tangent degree angle is equal to (opposite/adjacent)
	T					TANGENT) =
	o					(opposite,
	O					OPPOSITE)
	a					(adjacent,
	A					ADJACENT)		9tComputer

Remark: easy to remember, if pronounce like "So" "Ka" "Toe" (SOH CAH TOA) ... ;

( kankei Relations, RELATIONS, relations, Relations, relations), also see: Distance; k Usage; Mathematical Calculations; Mathematics; r Character;

( kyokusen Curve, Curve, Curve, Curve, Curve, Curve, Curve, Curve), also see: c Character Extension1 (e.g. Curve); curve; l Character (e.g. line); Mathematical Calculations (e.g. Integral); ( Water Clock: aroma, fragrance, odor, perfume, redolence, savor, smell);

Name	Mathematical Calculations	Remark
cos^-1 ((SQRT3) / 2)	i.e. arc cosine (??) degree	... for each pair of moon waves;
Sine x	= (O p p / H y p)	doko WHERE 0 < x < π ;
(a^m)^n = a^mn		Power M (Multiplex) Power N
a^mn = (a^m)^n
log a^m = m log a		Power to a Log
log (a by b)		Minus
log a by log b		Base
log (Base c) a b		Plus
log (Base a) a		variable As Base
a log (Base a) b		Distance to  b  (Walls)
a^(log (Base a) b) = b		a trick like 92x = 9x Dot Product  9x
		e.g. 10 (9 · 4) = 36 (10)

e.g. March 14, a.k.a. Pi Day (22/7 is 3.14, ((proportion, ratio, Radical102) of ( kyori Distance, kyori Distance, kyori Distance)) around the circle to the distance across it) ... ; Also see: Mathematics; Mathematical Calculations; p Character;

e.g. (Walls (a, b, c, x) Parameter (side-by-side Processor) : Teleportation ((SQRT2 (±98)) at Remote 100)) doko WHERE (, , ) ;

( log, log,  log, log) e.g. in Mathematical Calculations, log _{base can be any (Decimal, Fraction, Number)}, however ( , log 1 = 0, log _{(Decimal, Fraction, Number)} 1 = 0 ), i.e. very useful for Numerical values in numerology (log 1 is Zero), e.g. Numerlogical0 ... , And Then, "NOT log _{(Decimal, Fraction, Number)} 1" (mean, prompt) numerological (Val. , Ref. ) so numerological dimensional (e.g. sizes of "ro" ( , )), (numerological logical (e.g. LoShu magic square, Sudoku ( 3, 6, 9)), also see: Network Topology), numerological structural (e.g. (magic, plot, stratagem, trap, trick, Radical298), (measure, plan, plot, scheme, Radical731),  Scalar-Field (s), ( Table, Table, Table, Table)), ... ; Remark: log a.k.a. logarithm; ( log, log,  log, log);

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different of 2 cubes; e.g.

different of 2 cubes, i.e. a^3 - b^3 = (a - b) (a^2 + ab + b^2), a.k.a. a³ - b³ = (a - b) (a² + ab + b²), also see: c Character (cubes); 5cComputer; Distance (e.g. webp); Mathematical Calculations; Remark: in this case, think of 'a' as 1 cube; 'b' as 1 cube; webp is for (zoom-in, zoom-out) of 2 cubes; ( dozen, dozen, dozen, dozen), a.k.a. twelve, doko WHERE (x^2 - x^3 = 12), And Then, "square" formulary for each cube e.g. 6 surfaced DEE Box in Z-index ... ; "square" formulary e.g. (a² + b² = (a + b (a² - ab - b²))); "square" formulary e.g. (a² - b² = (a + b) (a - b)), ... ; Reminder: "square" is manmade, like cube is manmade;

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exponent; Exponent; e.g.

WHEN dealing with ( exponent, exponent, exponent, exponent, exponent), ( exponent, exponent, exponent, exponent, exponent), 1st to think of (Allow OR (NOT Allow)); if Allow, find WHICH ((( , , , ), ( , , , )) rule, Rule, RULE, rules, ... ) to apply, e.g.

don't confuse with (?^2 (square ones, ?^3 (cube ones));
don't confuse with Mathematical Operators, N - A R Y SUMMATION (" ∑ ") ;

Exponent, a.k.a. Power;
Exponents, a.k.a. Indices;

if Basic Latin, LATIN SMALL LETTER E ("e") ;
if Pre School English Typing .exe, ASCII ("101") ;
if Symbol, "e" in English language;
if Symbol, ESTIMATED SYMBOL (℮) ;
if Superscripts and Subscripts, ("ₑ") ;

Negative exponent indicates NOT multiplication;
Negative exponent means division (Method, Procedure, Technique);

regarding Mathematical Calculations, Log (at both sides) for solving exponents, if possible (Remark: sometime, impossible);
regarding Mathematical Calculations, can switch exponents, e.g. , i.e. (2^x)^3 = (2^3)^x ;

regarding Mathematical Calculations, can switch exponents, e.g. (2^y)^3 = (2^3)^y ;

regarding Mathematical Calculations, can switch exponents, e.g. (2^z)^3 = (2^3)^z ;

Times (based Number) i.e. exponent;

WORMHOLE can connect 2 diff regions at the same Time ... (Reference Books, also read: Prof. Michio Kaku's PHYSICS books) ; can switch exponents, so Teleportation may begin i.e. ( cosmos, space, universe) ... ;

( Zero, One), related to exponents would be: "exponent 0" means 1, on the other hand, "exponent 1" means Number itself;

e.g. 2^9 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 ; e.g. 3^8 = 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 ;
e.g. 2^8 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 ; e.g. 3^7 = 3 * 3 * 3 * 3 * 3 * 3 * 3 ;
e.g. 2^7 = 2 * 2 * 2 * 2 * 2 * 2 * 2 ; e.g. 3^6 = 3 * 3 * 3 * 3 * 3 * 3 ;
e.g. 2^6 = 2 * 2 * 2 * 2 * 2 * 2 ; e.g. 3^5 = 3 * 3 * 3 * 3 * 3 ;
e.g. 2^5 = 2 * 2 * 2 * 2 * 2 ; e.g. 3^4 = 3 * 3 * 3 * 3 ;
e.g. 2^4 = 2 * 2 * 2 * 2 ; e.g. 3^3 = 3 * 3 * 3 ;
e.g. 2^3 = 2 * 2 * 2 ; e.g. 3^2 = 3 * 3 ;
e.g. 2^2 = 2 * 2 ; e.g. 3^1 = 3 ;
e.g. 2^1 = 2 ; e.g. 3^0 = 1 ;
e.g. 2^0 = 1 ;

( exponent, exponent, exponent, exponent, exponent), ( exponent, exponent, exponent, exponent, exponent), also see: e Character; 5eComputer; Mathematical Calculations;

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factorial; Factorial; FACTORIAL; e.g.

IFF Command ( line, line, line, line) Parameter, And Then, type (e.g. C:\IT\ExecutableApplication\PreSchoolEnglishTyping.exe ) And Then, type "!," And Then, realize & understand THAT ASCII numerical value of the key is 33, WHICH has been used for (cheerful, grateful, wonderful, ... ) sign & symbol in English language, if compare to (factorial, Factorial, FACTORIAL) in Mathematics (Mathematical Calculations) ... e.g.

9! (mean (s), prompt (s), represent (s)) 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1;
8! (mean (s), prompt (s), represent (s)) 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1;
7! (mean (s), prompt (s), represent (s)) 7 * 6 * 5 * 4 * 3 * 2 * 1;
6! (mean (s), prompt (s), represent (s)) 6 * 5 * 4 * 3 * 2 * 1;
5! (mean (s), prompt (s), represent (s)) 5 * 4 * 3 * 2 * 1;
4! (mean (s), prompt (s), represent (s)) 4 * 3 * 2 * 1;
3! (mean (s), prompt (s), represent (s)) 3 * 2 * 1;
2! (mean (s), prompt (s), represent (s)) 2 * 1;
1! (mean (s), prompt (s), represent (s)) 1 * 1;
Remark: numerological linear calculations may begin ... ; numerological matrix (in common, (3 columns, 3 rows) calculations may begin ... ;

Binomial Coefficient e.g. (⁵₃) doko WHERE 5 (choose, pick, select) 3, other wordings would be:

n (choose, pick, select) k, calculated by fraction, e.g. take n 1st, And Then, apply formula

Binomial Coefficient (ⁿ_k) is equal to n! by (k! . (n-k)!), a.k.a. (ⁿ_k) = n! / (k! . (n-k)!) ... ;

in Computer Science . Browser, think of incoming Textbox (Val. , Ref. ) As FE (Front End), in this case, area of ". (n-k)!" ;

IFF Idea Processor (Buddha's Abhidharma based for humanoid nama) so k should be 5 then;

IFF Artificial Intelligence (AAI, AI, A I A) so k should be 4 then;

IFF Search Engine (3 lines are required) so k should be 3 then;

IFF 1&1 design model, 2! with 1! would be very useful, regarding (factorial, Factorial, FACTORIAL) ... ;

Also see: 3bComputer; 5cComputer;

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humanoid Jiwaka; e.g.

humanoid Jiwaka's chiryoyaku Therapeutic Gene Therapy System (kuru kuru WHILE commuting) with kadosei Mobility, by Gravity Dimension Computer ( directional gravity spot for each wheel ((e.g. AI Jumping like B Y D, originally invented by China in approx. since 2020s in our earth), e.g. Maglev Trains' Guided Track (Curves), e.g. Third WORMHOLE ( Elevating, Landing) for safety, sooth, and stability) ) ; Remark: Third WORMHOLE is beyond 90 degree to the defined USB (Suggestion: start with 1 meter diameter environment to the defined USB) a.k.a. 180 degree to the defined USB ( Walls alike open air, Walls alike O2 rich "human beings livable moons" in our universes, Walls like underwater); beyond (Natural Water Elevator (originally invented by Japan in approx. since 2010s) + Yellowish Variation) Method, Procedure, Technique, also see: (Analog Clock OR 時Time Ticks); Bright; Mathematical Calculations; Vehicle; I'm proud of China's new inventions ... ; Remark: I like to promote Korea's new inventions (e.g. 2020s designed modeled side-by-side Processor for Curved Displays) ... ;

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integral; Integral; INTEGRAL; e.g.

WHEN, doko WHERE, kuru kuru WHILE calculating a Curve, Integral is ((calculation based on something (heard, read) Radical951), deployed, implemented, used, utilized) with Limit (e.g. from minus to plus, from zero to infinity, from zero to one (a.k.a. decimal)) WHEN, WHERE, WHILE calculating a curve, based on a ( line, line, line, line) line e.g. (a line on x), ( a line on y), ( a line on z) ... regarding (SOH, CAH, TOA), Complexity Region (s) exist (if the line is a chord, if the line is a radius (i.e. related to radian), if the line is a tangent), ... , And Then, don't forget R&D yourself by (Applying, Using) Circle Rules a.k.a. Circular Geometric Theorem ... And Then, if from zero to one "decimal numbers" belong to (SQRT3, SQRT2), And Then, floating points may begin ... so (( Integral Upper Ref (TOP HALF INTEGRAL ("⌠"))), ( Integral Lower Ref (BOTTOM HALF INTEGRAL ("⌡")))) in Mathematical Calculations may (become, initialize) aesthetics in philosophy ... ;

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( INV, INV, INV, INV), inverse function, a function THAT 's reversing another function;

e.g. ( DEG . cos) 's INV is cos^-1;
e.g. ( DEG . e) 's INV is ( R A D . e);
e.g. ( DEG . ln) 's INV is e^x;
e.g. ( DEG . log) 's INV is 10^x;
e.g. ( DEG . sin) 's INV is sin^-1;
e.g. ( DEG . tan) 's INV is tan^-1;
e.g. ( DEG . !) 's INV is ( R A D . !);
e.g. ( DEG . π) 's INV is ( R A D . π);
e.g. ( DEG . ^) 's INV is ( R A D . ^);
e.g. ( DEG . "(") 's INV is ( R A D . "(");
e.g. ( DEG . ")") 's INV is ( R A D . ")");
e.g. ( DEG . √) 's INV is x^2;

INV is INVERSE, i.e. math usage ... ; Also see: Mathematics;
INV is INVOICE, i.e. business usage ... ; Also see: Biz;

I wrote: I've created a picture i.e. INVERSE, and I've a quiz: regarding (our earth, our moon), if I'm on the moon, HOW would the "INVERSE" would be?

well trained kids! replied: very easy quiz; since (area, environment) has lots of (Silver), Blue and Green are reversing each other sir;

I wrote: yes. so desu nal ... ; don't forget THAT (Yellow) and (Red) are also reversing ... ;

( INV, INV, INV, INV), also see: 9iComputer; Mathematical Calculations; Physics Law 181;

inverse cosine; e.g.

inverse cosine, a.k.a. arccos, denoted as:

	cos^-1	(		x	)
		OR	arccos	(		X	)
	cos^-1	(		y	)
		OR	arccos	(		Y	)
	cos^-1	(		z	)
		OR	arccos	(		Z	)
	cos^-1	(		x	)
		NOT	(	1	/	cos (x)	)
	cos^-1	(		y	)
		NOT	(	1	/	cos (y)	)
	cos^-1	(		z	)
		NOT	(	1	/	cos (z)	)

kuru kuru WHILE (Input (angle), Output (ratio)) OR

arccos ((Input (ratio e.g. DOMAIN [-1, +1])) OR

output (corresponding angle)) :

if DAL, also see: Schematic Symbols e.g. ;

if Definition, e.g. y = arccos (x), And Then, cos (y) = x;

if ( derivative, derivative, derivative, derivative) derivatives, (arccos x) d/dx = - 1 / √(1-x^2) doko WHERE x is (-1<x<1);

if DOMAIN, [-1, +1];

if Integral, ; Remark: numbers of "C" can be Capacitance in hardware;

if Range, a.k.a. Principal Value (Val.), e.g. [0, π] radian OR [0, 180] degree, so, output can be (indicated, limited, restricted) by range;

if Replication function, ID should obey (, arcsine (x) + arccos (x) = π divided by 2);

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Ln; Ln, a.k.a. Logarithm for base e ... ; e.g.

(( , , , ) i.e. Ln (1) = 0); in Mathematical Calculations, (Ln (1) a.k.a. Logarithm (1)) has been excluded, e.g. Keyword (nil, NULL) in Computer Science are neither Negative nor Positive (Also see: 8lComputer); other wordings would be NOT calculating Ln (1), however i.e. very useful in numerological, e.g. Numerological0, because non-zero (factors, floating points (SQRT3), floating points (SQRT2), numerical numbers, numerological mapped numbers) can be PCS based computing (since 2025) in addition to ZCS based computing, ... ; Reminder: because of PCS based computing, in the near future, ware ware We'll have (aroma, fragrance, odor, perfume, redolence, savor, smell), kuru kuru WHILE watching movie ... ; Remark: sayana (Stage, State, Status) is very important "New Usage," regarding neither Negative nor Positive ;

you may R&D yourself, e.g. kuru kuru WHILE calculating derivatives, if integer whole numbers prompt, then replace with zero ... ;
you may R&D yourself, e.g. kuru kuru WHILE calculating Logarithm, if Ln (1) prompt, then replace with zero ... ;
you may R&D yourself, e.g. kuru kuru WHILE calculating x, exponents of derivatives x should be excluded (nil, NULL);
you may R&D yourself, e.g. kuru kuru WHILE calculating y, exponents of derivatives y should be excluded (nil, NULL);
you may R&D yourself, e.g. kuru kuru WHILE calculating z, exponents of derivatives z should be excluded (nil, NULL);

if DAL, don't forget THAT is equal to Ln , also see: Schematic Symbols ( Y = Ln X), a.k.a. e constant, ... ;

Reminder: kuru kuru WHILE calculating (Kilo, Mega, Giga, Tera, Peta, ... ) by Multiplier, be careful with whether "2 to the power" or "10 to the power," WHEN (ln, logarithm (log)) for calculating very large numbers ... ; in common, "ln" is more likely to be with Eccentric _circles ... ;

Logarithm, a.k.a. ( log, log, log, log, log), 'Inverse Function of Exponential Function' is called Logarithm, a.k.a. log : e.g. users (large number of (Personal Directories, Desktop Directories)), log-in, log-out, in Network Topology, by Logarithm ... ; e.g. (Squared vs. Square Root), doko WHERE Inverse Function (s) are essential, by Logarithm ... ; e.g. scale of moon waves (directional gravity spots (Gravity Dimension Computer)), by Logarithm ... ; e.g. log base (s), good example would be: (log₃ vs. log₂), (log₂ vs. log₁₀) ... ; e.g. Keyword (s) : (do, undo), ((Input (Val., Ref.) <> Inverse of the Input) And Then, result Output (Val., Ref.) should be the same as Input (Val., Ref.) [think of User Name and Password to be allowed (if correct) at Input Function, And Then, the user is at "Inverse of the Input" (Stage, State, Status) to a system, And Then, the user is no longer using the system, so Output Function (keeps, maintains) the same User Name and Password] ),  logarithm can be used to solve exponential, (log-in, log-out), by Logarithm ... ; e.g. Hydrogen minus ion (pH ( Water)), Gas Level R&D in Nanobot Programming, by Logarithm ... ; e.g. (Cycle, Frequency, Wave), also see: Radical514 of (e.g. the defined device's amplifier, e.g. the defined iroLED 's from points of line to its haba Breadth), by Logarithm ... ; if Number Base ( b), and Number ( N) are positive, kuru kuru WHILE Base ( b) is NOT equal to one (1), And Then, ((Logarithm Base b) to the Number (N) is equal to k), e.g. log _b N = k, ((Logarithm Base b) to the Number (N) is equal to k) IFF ("if and only if") (Base b to the power ('Exponential') k is equal to the Number (N)), b ^k = N, (Base b to the power ('Exponential') k is equal to the Number (N)) i.e. Logarithm, a.k.a. ( log, log, log, log, log) ... ; Reminder: base is not equal to one (1) because, dealing with curves, ... ; "k" is somehow related to (heat, temperature), ... ; good example would be log₁₀ (100) = 2;

, i.e. x is equal to log based (Subscript a) to y AND y is equal to a to the power (exponential) x, a.k.a. ( x = log _a y) AND ( y = a^x), recommended to measure (Anti-Earthquake (Japan Meteorological Agency (J M A) Seismic Intensity Scale, Richter Scale earthquake intensity, Shindo Scale earthquake magnitude), data (analyzing linear exponentials), (exponential decay, exponential growth), rate, sound intensity (dB), (-time, Time), Water (pH Level in Chemistry)); Remark: since AND is at the midst, Left's x (Val., Ref.) should be TRUE, and Right's y (Val., Ref.) should be TRUE also;

base two Logarithm (e.g. log₂), a.k.a. Binary Log in Computer Science ... ;
base ten Logarithm (e.g. log₁₀), a.k.a. Natural Logarithm ... ;
Ln, a.k.a. Logarithm for base e ... ;
log (0, zero) is undefined (e.g. Syntax ERROR); log (1, one) is (0, zero); Also see: -log; Mathematical Calculations; Numerological0;

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Ordinary Differential Equations (ODE) e.g.

( ODE, ODE, ODE, ODE), Ordinary Differential Equations (ODE) i.e. NOT partial differential equations, e.g. WHAT is ODE? Answer would be among derivatives, form (s), (((measure, plan, plot, scheme, Radical731), (magic, plot, stratagem, trap, trick, Radical298)) plot (s)), (( Table, Table, Table, Table) table (s)), Scalar-Field (s), e.g.

small amount of ( x) would be 'dx' ... ,
small amount of ( y) would be 'dy' ... ,
small amount of ( z) would be 'dz' ... ,

regarding Basic (x, X), if ( interval, Real-time, timeslot) are needed in Mathematics, e.g. ( x + 3 = 2^2 ) can be represented as 3x/dt + 3 = 2^2, so instead of "x," its derivatives (in this case: dx/dt) doko WHERE t (-time, Time) is, so ODE can be realized & understood; another example in (2 * 6) momentum (our earth) would be Newton's (F = ma) doko WHERE acceleration "a" can be 2nd derivatives (e.g. a = (dv (Velocity)) / dt (Time)) = 2nd derivatives on "x" (d^2 x / d t^2 (-time)) WHERE "m" is a Number AND 2nd derivatives of its ( position, position, position, position) "a", also see: 3; 2; Mathematical Calculations; 6oComputer; by the way, in Computer Science, if you know five e.g. (Left's 5, Right's 5), And Then, you should be global military generals (if nothing to do with money), OR you should be billionaires of US Dollar rich (if something to do with money (e.g. donations, gifts, ... )), and don't forget THAT KNOWLEDGE IS POWER, starting from e.g. the most challenge in 21st century & beyond would be Anti-Earthquake design model, Gene Therapy System (fully Recovery to WHOM) by Nanobot Programming, Natural Water Elevator (kadosei Mobility, PCS based computing in addition to ZCS based computing (since 2025), ZCS based computing), ... ;

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quadratic; e.g.

kuru kuru WHILE dealing with squared x, (e.g. x^2),
kuru kuru WHILE dealing with squared y, (e.g. y^2),
kuru kuru WHILE dealing with squared z, (e.g. z^2),

quadratic (General Form, general term) e.g. (ax^2 + bx + c = 0) a.k.a. (ax² + bx + c = 0) ... ; Remark: kuru kuru WHILE factoring, in common, 4ac has been used in Quadratic Formula ... ; Reminder: apply to (x^2, y^2, z^2) squared ones only AND (NOT For (x^3, y^3, z^3)) ... ;

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radian; Radian; RADIAN; e.g.

Remark: in Computer Science ( cosmos, space, universe), don't confuse between ( 15 degree by SQRT2 longitudes) and ((SQRT15)/2), and recommended formulary for Mathematical Calculations, e.g.

, i.e. (x - a)^2 + ( y - b)^2 = r^2 As Radius; Also see: radian;
, i.e. a^2 + 2ab + b^2 = (a + b)^2 As Pythagorean;
, i.e. Distance (d) = √((x₁ - x₂)^2 + (y₁ - y₂)^2) As Hypotenuse;

, i.e. (f (x, y) = ((x^2) + (y^2)) = 13 WHEN ((x = 3) and (y = 2))), a.k.a. f (x, y) = ((x^2) + (y^2)) Output 13 WHEN (( x=3) AND ( y=2)), a.k.a. FUNCTION OF ( X, Y) is equal to 'X SQUARE' plus 'Y SQUARE' ; Remark: doko WHERE 500.98+ would be: remote ((0.5), (1/2), (1:2)), a.k.a. Half Way (Method, Procedure, Technique) in Network Topology ... ; kuru kuru WHILE dealing with many ( derivative, derivative, derivative, derivative) derivatives e.g. multi-systems ( Time) should be 1st priority, e.g. nitotte For DOMAIN [-1, +1], Laplace Transform (Method, Procedure, Technique) should be calculated, deployed, implemented, measured, used, utilized, ... ; Also see: 3; 2; Mathematical Calculations; radian; since 1980s, multi-systems inventor Toshiba has been very very very genius ... ;

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rule; Rule; RULE; rules; e.g.

(( , , , ), ( , , , )) rule, Rule, RULE, rules, ... ;

Chain Rule, if Inner Function AND Outer Function are together ... ; e.g. (Inner derivative . Outer derivative) ... ; Remark: Automata Theory books are recommended to read, if in-depth logics of (inner, outer) ... ;

Constant Rule, e.g. ƒ (x) = 5, And Then, its derivative ƒ '(x) = zero, ... ; Remark: very useful in numerological because of computers need (nil, NIL, nothing, Null, NULL) ... ;

Constant Rule, e.g. ƒ (x) = -√2, And Then, its derivative ƒ '(x) = zero because, there is no "x" in Right Side of the equation " = ";

Constant Rule, e.g. ƒ (x) = exponent e¹⁰⁸, And Then, its derivative ƒ '(x) = zero because, there is no "x" in Right Side of the equation " = ";

(Difference (Minus (" - ")), Sum (Plus (" + "))) Rule, e.g. adding (Plus OR Minus) to the (Left AND Right) of equation (" = ") ... ; Remark: very useful in NAND gates (Hardware, Software), in addition to Swap Symbol (x^=y^=x^=y, formulated WHILE studying MS in CS Architecture homework assignment in 1995) ... ;

(Multi- , Multiple) Rule (e.g. constant's derivative is zero i.e. mathematical approach) ... ;
(Multi- , Multiple) Rule (e.g. using multiplexor, also see: Multiplier, i.e. computational approach) ... ;
(Multi- , Multiple) Rule (numerological dimensional needs zero, for computational (nil, NIL, nothing, Null, NULL)) ... ;

Power Rule, e.g. if (?)⁷ then 7(?)⁶ ; its derivative would be 1000(?)⁹⁹⁹ if ƒ (?)¹⁰⁰⁰ And Then its derivatives ... ;

Product Rule, e.g. e.g. ƒ (x) = x³ . x² = x³⁺² = x⁵, And Then, if (x³ As u), (x² As v), And Then, its derivative ƒ '(x) = u' . v + u . v' ; Also see: Mathematical Calculations; 7pComputer; Products;

Quotient Rule, e.g. cos^-1 (SQRT3/2) nitotte For kaku Each pair of moon waves in degree, And Then, Quotient Rule for fraction, e.g. Quotient Rule for fraction ƒ (z) = u ÷ v , And Then, it derivative ƒ ' (z) = ( u ' . v - u . v ' ) ÷ v^2; Also see: 3; 2; Anti-Earthquake; d Computer (derivative); Mathematical Calculations; area between curves, 1st. intersecting points (e.g. 1 chord of 2 circles); 2nd. using Absolute Value (a.k.a. Val. ) Bar i.e. using "|     |" e.g.

 | _a ^b ƒ (x) - g (x) dx | doko WHERE 'g' for Gravity;

 3rd. solve integral; 4th. anti-derivative; 5th. years ahead, in our earth, NOT EXIST (Earthquake) prompt means the Shakya King will GRANT DEED (Me, Me, Me, Me) 1 whole human beings livable moon among many human beings livable moons in our universes ... ;

well trained kids!! replied: HOW can we calculate to solve integral sir?
I wrote: OH!! OH!! You need to realize & understand 180 degree to the defined USB, good example would be: WHEN very very fast trains are traveling at a curve, to maintain (on tracks + its speed) means dubbing DEE spot at 180 degree to the defined USB, so results of ( Elevating, Landing) for each side (of the very very fast trains) can be calculated by Processor , so called "Side-by-Side Processor" ... ; Remark: this info is worth more than trillions of US Dollar; good example would be, e.g. till 2026, I still cannot dub My 2024 Toyota Land cruiser 1958 Edition's Front hikari Light ("Left and Right to be together at the front means already dubbed DEE at 180 degree to the defined USB") ... ; Also see: c Computer ( curve ( kyokusen Curve, Curve, Curve, Curve, Curve, Curve, Curve, Curve)); Maglev Train; Mathematical Calculations;

You should have your own rules ... e.g. Your Own Rule ... ; You can have your own rule e.g. Purple.Green.Radio.Mapping (for our earth only), OH+ OH+ My Goodness!! military grade has begun, e.g. 2+2 designed modeled 2 M1A1H17Titanium22 tanks + 2 humanoids  are unbeatable && unbreakable ; Me too sir, I'm holding my airborne Nanobot (s) in Nanobot Programming (NOT to have COVID-19 alike pandemic) ... ; Me too sir, I'm holding My Manmade Global Weather (the defined regions, NOT to have bad weather conditions) ... ;  OH+ OH+ My Goodness!! I need a cave to hide-out, or Laser Shields to my living environments ... ;

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sector; Sector; e.g.

Area of Sector, i.e. (π r^2 . x /360), a.k.a. (π r² . x/360), also see: a Character (Area); Eccentric _circles; Mathematical Calculations (Area of Sector); sCharacterExtension1 (Sector);

term; e.g.

( term, term, term, term, term) in (Arithmetic Series, Geometric Series), if average (AVG) e.g. AVG of Serial (Series) Number is equal to the Number dot (term (t) i + t Number) by 2, a.k.a. Series (S) Number (n) Sn = n . (term (t)i + tn) /2; Also see: approx variable DEE Patterns; iroLED; Mathematical Calculations; t Character; 2tComputer; variable;

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Also see: method; operator;

2008 Fuzzy mathematical calculation, (also see: http://documents . wolf ram .com / applications / fuzzy logic / Manual /10 .html), so that addition, subtraction, multiplication, image, max function, min function, ... can be further studied in fuzzy approach ... ;

2007 ADC's distortion causes its sample rate becomes (sample rate / 2), and bandwidth signal [i.e. S N R 50dB at 12bit per 1G sample per second] prompts, and then processing gain can be calculated as:
processing gain = -10(log(bandwidth_signal/(sample rate/2))).
If jitter, for the given system, S N R to jitter calculated as:
S N R is approximately -20(log₁₀((2π_cycle f _{frequency _input}) (t _jitter))) ;

2007 I P M motor's Torque: Depending on types of motor, calculation may vary; In I P M motor [e.g. air con compressor motor], when calculating magnet _torque and reluctance _torque, 2 * rotor angle in sine is used as:
torque T = P _n ((Φ * ampere I * (cosine (β))) + (1/2(inductance L _q - inductance L_d) * (square (ampere I)) ) * sine(2 * β)); AC input into rectifier, and then normal power supply P _n can be measured. The given F O C is sensor-less. Velocity control can be further studied.  IFF 2 phase circuit, while current I is looping, flux can be estimated, because the circuit's back E M F are in cosine and sine as: volt V₁ = (resistance R _s * current I₁) + (inductance L_s * (dI₁ / d t vary I1 at time t)) + (d / d t at time t(-Ψ _r * cosine (θ _r))); V₂ = (resistance R _s * current I₂) + (inductance L_s * (dI₂/d t vary I2 at time t)) + (d / d t at time t(-Ψ _r * sine (θ _r))); WHERE rotor's flux angle in cosine and sine are as: Ψ _r * cosine (θ _r) = integration ∫ integral in distance as ∫((resistance R _s * current I₁) - V₁) + (inductance L_s * I₁); Ψ _r * sine (θ _r) = ∫((resistance R _s * current I₂) - V₂) + (inductance L_s * I₂);

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Basic Calculator ... ;

(equation, expression, form, formula, function, method, system) Radical992;

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