Principle Library

Deep dives into specific concepts. Each principle guide covers derivations, applications, and common pitfalls.

149 principles across 2 domains

Mathematics

Algebra (30)

Absolute Value - Definition: The Piecewise Distance Rule

Absolute Value Cases: Split Into Two Branches

Additive Equality: Adding the Same Quantity to Both Sides

Additive Inequality: Same Quantity, Both Sides

Additive Inverse: Canceling an Offset with Its Opposite

Clear Denominators: Multiply to Remove Fractions from Equations

Combine Like Terms: Merge Coefficients of Identical Powers

Completing the Square (Rewrite Identity): Quadratic to Vertex Form by Pattern Substitution

Cross Multiplication: Solving Proportions by Cross Products

Difference of Squares: Factor a Squared Difference into Two Binomials

Distributive Property: Expand Multiplication Over Addition

Exponent Power Rule: Multiply Exponents in a Nested Power

Exponent Product Rule: Add Exponents When Multiplying Same-Base Powers

Exponential Model: Equal Multiplier per Unit Input

Factor Common Term: Reverse the Distributive Property

Linear Model: Understanding Slope and Intercept

Multiplicative Equality: Multiply or Divide Both Sides

Multiplicative Inequality: Flipping When the Multiplier Is Negative

Multiplicative Inverse: Canceling a Factor with a Reciprocal

Negative Exponent Rule: Rewrite to Reciprocal Form

Perfect Square Trinomial: Rewrite as a Squared Binomial

Proportional Model: Mastering Direct Variation with y = kx

Quadratic Factoring: Rewrite a Quadratic as a Product of Linear Factors

Quadratic Formula: Extract Both Roots Without Factoring

Quadratic Model: Parabolic Curves and Second-Degree Structure

Radical Definition: The Principal Nonnegative Root Rule

Simplify Radicals (Square Factor): Extract Perfect-Square Roots

Square Root Property: Splitting a Squared Equation into Two Branches

Substitution Property: Replace Any Expression with an Equal One

Zero Product Property: From Factored Form to Solutions

Calculus (47)

Antiderivative definition: Recognizing the Reverse Derivative

Average value of a function: Integrating to Find the Mean

Chain rule: Differentiating Composite Functions

Continuity at a Point: Definition and the Three-Part Test

Definite integral (Riemann sum form): Area as a Limit of Sums

Derivative at a Point (Definition): The Difference Quotient

Derivative Constant Multiple Rule: Factor Constants Out of Derivatives

Derivative of a constant: Any constant differentiates to zero

Derivative of a^x: Why the Log Factor Appears

Derivative of e^x: The Function Equal to Its Own Derivative

Derivative of ln(x): Differentiate the Natural Log on Its Domain

Derivative sum rule: Distribute differentiation over addition

Differentiate both sides: Preserve equality under differentiation

First derivative test (local maximum): Infer a local peak from a positive-to-negative sign change

First derivative test (local minimum): Infer a local minimum from a sign chart

Fundamental Theorem of Calculus (Part 1): Differentiate accumulation integrals directly

Fundamental Theorem of Calculus (Part 2): Evaluate a definite integral from an antiderivative

Indefinite Integral as Antiderivative: Finding Every F with F' = f

Integral constant multiple rule: Pull constants outside an integral

Integral of 1/x: Recognize the Logarithmic Antiderivative Away From Zero

Integral of a^x: Why the Log Divisor Appears

Integral of e^x: Antidifferentiate the natural exponential in one step

Integral sum rule: Split an integral before integrating each part

Integration by parts: Reassign derivative and antiderivative roles in a product

L'Hopital's rule: Replace an indeterminate quotient with a ratio of derivatives

Left-hand limit statement: f(x) Approaching L from the Left

Limit constant multiple rule: Pull a constant factor out of a limit

Limit of a Constant: Evaluating Constant Limits Directly

Limit of a continuous composition: Push the limit through an outer function

Limit of the Identity: Direct Substitution Base Case

Limit product rule: Split a product to evaluate limits separately

Limit quotient rule: Split a quotient to evaluate limits separately

Limit statement: What It Means for f(x) to Approach L

Limit sum rule: Split a sum to evaluate limits separately

Linear Approximation: Estimating Function Values Near a Point

Power rule for integrals: Integrate x^n when n is not -1

Power rule: Differentiate any integer power of x

Product Rule: Differentiating Products of Functions

Quotient Rule: Differentiating Ratios of Functions

Right-hand limit statement: f(x) Approaching L from the Right

Second derivative test (local maximum): Classify a critical point from concavity

Second derivative test (local minimum): Infer a local minimum from curvature at a critical point

Squeeze theorem: Evaluating limits by bounding

Substitution rule (definite integral): Change variables and bounds together

Substitution rule (u-substitution): Rewrite integrals by changing variables

Tangent Line Equation: Slope from the Derivative

Taylor Polynomial Definition: Approximating Functions with Polynomials

Functions (12)

Affine Transform Form: Parameters of a Transformed Function

Apply Inverse to Both Sides: Isolate a Variable Inside a Function

Composition Definition: Feeding One Function's Output into Another

Composition Expansion: Evaluate a Composed Function in Two Steps

Evaluate by Substitution: Plug an Input into a Function Rule

Function Rule Definition: What It Means to Define a Function by a Rule

Inverse Cancellation: Recover the Original Input or Output

Inverse Definition: Reversing a Function's Input and Output

Log-Exponential Rewrite: Convert Between Log and Exponential Forms

Logarithm Model: The Inverse Relationship to Exponentiation

Piecewise Branch Selection: Evaluating the Right Rule

Piecewise Definition: Define Functions with Multiple Rules Across Regions

Physics

Classical Mechanics (60)

Acceleration - Derivative Definition: Instantaneous Rate of Change

Angular Acceleration - Derivative Definition: Instantaneous Rotational Change

Angular Displacement Integral Relation

Angular Impulse Formula: When Torque Over Time Changes Angular Momentum

Angular Impulse-Angular Momentum Theorem (Integral): Time-Varying Torque

Angular Momentum (Particle): Rotational Motion About a Point

Angular Momentum of a Rigid Body: Fixed-Axis L = Iω

Angular Velocity - Derivative Definition: How Rotation Rate Changes

Angular Velocity Change: Integral Form for Rotational Kinematics

Arc Length-Angle Relation: Connecting Linear and Rotational Motion

Center of Mass (Position): Locating the Balance Point

Center of Mass (Velocity): Finding the Average Motion

Centripetal Acceleration: Understanding Acceleration in Circular Motion

Conservation of Angular Momentum

Conservation of Linear Momentum

Displacement from Average Velocity — Kinematics 4

Displacement-Integral Relation: Finding Position from Velocity

Gravitational Potential (Near Surface): Mastering the mgh Formula

Hooke's Law: Understanding Spring Force and Elastic Behavior

Impulse-Momentum Theorem (Algebraic): Impulse Equals Momentum Change

Impulse-Momentum Theorem (Integral): General Time-Varying Forces

Kinematics 1 - Algebraic: Predicting Position Under Constant Acceleration

Kinematics 2 - Algebraic: Velocity as a Function of Time

Kinetic Friction: Understanding Sliding Resistance in Mechanics

Linear Momentum (Definition): Quantifying Motion's Persistence

Mechanical Energy Conservation: Master Energy Transformations

Mechanical Energy with Non-Conservative Work: Tracking Energy Transfers

Moment of Inertia - Integral Definition: Continuous Mass Distributions

Moment of Inertia (Discrete): Quantifying Rotational Inertia

Newton's First Law (Rotation): Rotational Equilibrium

Newton's First Law (Translation): When Forces Balance

Newton's Law of Gravitation: Understanding Universal Attraction

Newton's Second Law - Momentum Form: Force as Rate of Change

Newton's Second Law (Rotation): Connecting Torque to Angular Acceleration

Newton's Second Law (Translation): The Foundation of Force and Motion

Newton's Third Law: Identify Action–Reaction Pairs

Parallel Axis Theorem: Shifting Rotational Inertia

Power - Derivative Form: Instantaneous Rate of Energy Transfer

Power (Algebraic): Calculating Energy Transfer Rate

Power (Rotation): Instantaneous Rate of Rotational Energy Transfer

Rotational Kinematics 1: Angular Position Under Constant Angular Acceleration

Rotational Kinematics 2: Angular Velocity with Constant Angular Acceleration

Rotational Kinematics 3: Angular Velocity Without Time

Rotational Kinematics 4: Angular Displacement via Average Angular Velocity

Rotational Kinetic Energy: Master Energy in Rotating Systems

Rotational Work: Energy from Constant Net Torque

Spring Potential Energy: Elastic Energy Storage

Static Friction: Understanding the Force that Prevents Sliding

Tangential Acceleration: Connecting Linear and Rotational Motion

Tangential Speed: Definition, Formula, and When to Use v = rω

Torque - Angular Momentum Form: Rotational Dynamics in Calculus Form

Torque (Definition): Measuring Rotational Force Effectiveness

Translational Kinetic Energy: Understanding Motion's Energy

Translational Work: Energy from Constant Force

Velocity - Derivative Definition: Instantaneous Rate of Position Change

Velocity Change - Integral Relation: Find Velocity from Acceleration

Velocity-Position Relation: Kinematics 3

Weight (Near Surface): Understanding Gravitational Force in Local Contexts

Work - Integral Definition: Computing Work Along Arbitrary Paths

Work-Energy Theorem: Connect Force to Speed

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