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34 results for “rule”

Formulae (32)

Rounding Rule

5\text{ or more} \uparrow, \ 4\text{ or less} \downarrow

Round up if the next digit is 5+, otherwise round down.

AlgebraIntermediate

Logarithm Product Rule

\log_b(xy) = \log_b(x) + \log_b(y)

The logarithm of a product equals the sum of logarithms.

AlgebraIntermediate

Logarithm Quotient Rule

\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

The logarithm of a quotient equals the difference of logarithms.

AlgebraIntermediate

Logarithm Power Rule

\log_b(x^n) = n \cdot \log_b(x)

The logarithm of a power equals the exponent times the logarithm.

AlgebraIntermediate

Logarithm Product Rule

\log_b(xy) = \log_b x + \log_b y

The log of a product is the sum of the logs.

AlgebraIntermediate

Logarithm Quotient Rule

\log_b\frac{x}{y} = \log_b x - \log_b y

The log of a quotient is the difference of the logs.

AlgebraIntermediate

Logarithm Power Rule

\log_b(x^n) = n\log_b x

The log of a power brings the exponent to the front.

AlgebraAdvanced

System of Two Linear Equations

x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}

Cramer-rule solution for two simultaneous equations.

CalculusIntermediate

Power Rule (Derivative)

\frac{d}{dx}x^n = n x^{n-1}

Derivative of a power function.

CalculusIntermediate

Chain Rule

\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)

Derivative of a composite function.

CalculusIntermediate

Product Rule

(fg)' = f'g + fg'

Derivative of a product of two functions.

CalculusIntermediate

Quotient Rule

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Derivative of a quotient of two functions.

CalculusIntermediate

Power Rule (Integral)

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \; n \ne -1

Antiderivative of a power function.

CalculusAdvanced

Integration by Parts

\int u\,dv = uv - \int v\,du

Integration technique from the product rule.

CalculusAdvanced

L'Hôpital's Rule

\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}

Evaluates indeterminate limits 0/0 or ∞/∞.

CalculusBasic

Constant Rule (Derivative)

\frac{d}{dx}(c) = 0

The derivative of any constant is zero.

CalculusBasic

Sum Rule (Derivative)

(f+g)' = f' + g'

The derivative of a sum is the sum of the derivatives.

CalculusBasic

Constant Multiple Rule

\frac{d}{dx}(c\,f) = c\,f'

Constants factor out of a derivative.

CalculusBasic

Power Rule

\frac{d}{dx}x^n = nx^{n-1}

Derivative of a power function.

CalculusIntermediate

Product Rule

(uv)' = u'v + uv'

Derivative of a product of two functions.

CalculusIntermediate

Quotient Rule

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Derivative of a quotient of two functions.

CalculusIntermediate

Chain Rule

\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}

Derivative of a composite function.

CalculusBasic

Power Rule for Integration

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Antiderivative of a power function (n≠-1).

Probability & CombinatoricsBasic

Complement Rule

P(A') = 1 - P(A)

Probability that an event does not occur.

Probability & CombinatoricsIntermediate

Addition Rule

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability that A or B occurs.

Probability & CombinatoricsBasic

Multiplication Rule (Independent)

P(A \cap B) = P(A) \cdot P(B)

Probability of both independent events occurring.

Aviation & AeronauticsBasic

True Airspeed (rule of thumb)

TAS \approx IAS + (IAS \times 0.02 \times \tfrac{alt}{1000})

Quick estimate: true airspeed increases about 2% per 1000 ft of altitude above indicated airspeed.

Finance & InterestBasic

Rule of 72

t \approx \frac{72}{r}

Approximate years to double an investment.

Time & CalendarBasic

Leap Year Rule

\text{leap} \iff (y \bmod 4 = 0 \;\wedge\; y \bmod 100 \neq 0) \;\vee\; (y \bmod 400 = 0)

Tests whether a Gregorian year has 366 days.

Number TheoryBasic

Divisibility Rule for 3

n \text{ divisible by } 3 \iff \text{digit sum divisible by } 3

A number is divisible by 3 if the sum of its digits is.

Number TheoryBasic

Divisibility Rule for 9

n \text{ divisible by } 9 \iff \text{digit sum divisible by } 9

A number is divisible by 9 if the sum of its digits is.

Number TheoryIntermediate

Collatz Step

n \to \begin{cases} n/2 & \text{even} \\ 3n+1 & \text{odd} \end{cases}

The rule of the famous unsolved Collatz conjecture.

Math Tricks (2)

Divisibility Quick Checks

Quickly check if a number is divisible by 2 through 12 without dividing.

View trick

Rule of 72 (Doubling Money)

Estimate how many years it takes an investment to double: divide 72 by the annual interest rate.

View trick