CHAPTER 04
Differentiation
Develop the derivative from the idea of change and understand what differentiation means before learning its rules and applications.
DIFFERENTIATION
From limits to derivatives
Derivative from First Principles
Build the derivative from the difference quotient and a limiting process, including its meaning as slope and instantaneous rate of change.
Learn → 02Derivative Notation
Understand \(f'(x)\), \(dy/dx\), differentiation operators and higher-derivative notation.
Learn → 03Differentials: \(dx\) and \(dy\)
Understand differentials, linear change and the relationship \(dy=f'(x)dx\).
Learn → 04The Power Rule
Differentiate powers and understand how the rule follows from the definition of the derivative.
Learn → 05Sum, Difference and Constant Rules
Differentiate sums, differences, constants and constant multiples.
Learn → 06The Product Rule
Differentiate expressions formed by multiplying functions.
Learn → 07The Quotient Rule
Differentiate expressions formed by dividing one function by another.
Learn → 08The Chain Rule
Differentiate composite functions and understand how rates of change combine.
Learn → 09Exponential and Logarithmic Derivatives
Differentiate exponential and logarithmic functions and combine their rules with the chain rule.
Learn → 10Trigonometric Derivatives
Differentiate sine, cosine and tangent functions, including composite forms.
Learn → 11Implicit Differentiation
Differentiate relationships where \(y\) is not written explicitly as a function of \(x\).
Learn → 12Higher-Order Derivatives
Understand second and higher derivatives, concavity and successive rates of change.
Learn → 13Stationary Points
Use derivatives to identify maxima, minima and stationary points of inflection.
Learn → 14Optimisation
Use derivatives to find values that maximise or minimise quantities.
Learn → 15Related Rates
Use the chain rule to connect rates at which related quantities change.
Learn →The central idea
Differentiation is fundamentally the study of instantaneous change. The differentiation rules are efficient consequences of this underlying limiting definition.