dy/dx
The Change Engine
Derivatives, Rules, Rates of Change & Optimization
The Central Question
World 17 asked: What does a quantity approach? Now we ask: How fast is it changing at this exact moment? A derivative measures how one quantity changes with respect to another. It converts a function into a description of how that function changes.
POSITION →
CHANGE →
AVERAGE CHANGE →
SMALLER INTERVAL →
LIMIT →
INSTANTANEOUS CHANGE →
DERIVATIVE →
SLOPE →
RATE →
OPTIMIZATION
The 63 Differential Cells
From Secant Lines to Optimization
CELL 01-06
Foundations of Change
- Change & Finite Difference
- Average Rate of Change
- The Secant Line
- The Tangent Question
- First Principles Definition
CELL 07-16
The Derivative
d/dx (xⁿ)
↓
n xⁿ⁻¹
- Derivative as Slope & Rate
- Notation (f', dy/dx)
- Derivative of Constant & x
- Power Rule & Sum/Difference
- Polynomial Derivatives
CELL 17-22
Differentiation Rules
f'g
+
fg'
- Product Rule & Interpretation
- Quotient Rule
- The Chain Rule
- Composition as Change Pipeline
CELL 23-27
Special Functions
eˣ → eˣ
- Exponential (eˣ and aˣ)
- Logarithmic (ln x)
- Trigonometric Functions
- Inverse Trigonometric Functions
CELL 28-32
Advanced Differentiation
- Implicit Differentiation
- Parametric Differentiation
- Second Derivative (f'')
- Higher Derivatives
- Differential Eq. Preview
CELL 33-39
Geometry of Change
- Tangent & Normal Lines
- Increasing / Decreasing
- Critical Points
- Concavity & Inflection Points
CELL 40-46
Optimization
- Local Maxima & Minima
- First & Second Derivative Tests
- Absolute Extrema
- Optimization with Constraints
CELL 47-53
Applications & Rates
- Related Rates in Systems
- Error & Approximation
- Differentials (dy = f'(x)dx)
- Physics, Econ, Bio, Chem Models
CELL 54-56
Theorems
- Rolle's Theorem
- Mean Value Theorem (MVT)
- Local vs Global Behavior
CELL 57-59
Frontiers
- Derivative as a New Function
- Machine Learning Preview
- Gradient Optimization Preview
CELL 60-63
The Change Engine Synthesis
d/dx
- Limit → Derivative
- Secant → Tangent
- Function → Rate
- Mastery of The Change Engine
Exit Portal
We have learned to measure exactly how fast a quantity is changing at any given instant. But this creates the great calculus mirror: If differentiation tells us how a quantity changes, can we reverse the process and recover the accumulated quantity?