Calculus
𝒅𝒚
Differential calculus 𝒅𝒙
For an introduction to differentiation:
• A brief refresher on basic differentiation, critical points and their nature, and with
applications to economics.
Introduction to calculus (pdf, 78KB)
• A more in-depth treatment to differentiation: rates of change, tangents and
derivatives, the product, quotient and chain rule, stationary points and optimisation
problems.
Introduction to differential calculus (pdf, 2.1MB)
For specific help on calculating derivatives using the rules of differentiation:
• Differentiating constants 𝑦 = 𝑘, polynomial functions 𝑦 = 𝑥 𝑛 , constant multiples 𝑦 =
𝑐𝑓(𝑥), addition and subtraction of functions 𝑦 = 𝑓(𝑥) ± 𝑔(𝑥), the product rule for
𝑢
𝑦 = 𝑢𝑣, and the quotient rule for 𝑦 = 𝑣 .
The rules of calculus (pdf, 89KB)
• The chain rule for composite functions 𝑦 = ℎ(𝑔(𝑥)), and its two formulations:
𝑑𝑦 𝑑𝑦 𝑑𝑢
𝑦 ′ = ℎ′ (𝑔(𝑥))𝑔′(𝑥) and = 𝑑𝑢 × 𝑑𝑥.
𝑑𝑥
Composite function rule (the chain rule) (pdf, 88KB)
• For derivatives of functions with exponentials 𝑒 𝑥 and logarithms ln 𝑥. Some of the
examples assumes knowledge of the chain rule.
Derivatives of exponential and logarithmic functions (pdf, 81KB)
• For derivatives of functions with sin 𝑥 , cos 𝑥 , tan 𝑥. Some of the examples assumes
knowledge of the product, quotient and chain rules.
Differentiation of trigonometry (pdf, 78KB)
For what derivatives can tell us about the shape of a graph:
𝑑𝑦
• The first derivative 𝑑𝑥 : increasing and decreasing functions, stationary points and their
nature, relative maximum and minimum.
The first derivative and stationary points (pdf, 99KB)
𝑑2 𝑦
• The second derivative 𝑑𝑥 2: concave up, concave down and points of inflection.
Second derivative and points of inflection (pdf, 95KB)
Integral calculus ∫ 𝒇(𝒙)𝒅𝒙
For an introduction to the indefinite integral ∫ 𝑓(𝑥)𝑑𝑥: anti-derivatives, calculating some
elementary anti-derivatives and reversing the Chain Rule.
Introduction to integration part 1: the anti-derivative (pdf, 191KB)
𝑏
For an introduction to the definite integral ∫𝑎 𝑓(𝑥)𝑑𝑥: limiting sums, the Fundamental Theorem
of Calculus, and finding areas under and between curves.
Introduction to integration part 2: the definite integral (pdf, 281KB)
Applications of calculus
For demonstrations of how to use the concepts and tools of differential calculus to sketch
graphs and curves of functions through several worked examples.
Curve sketching using calculus (pdf, 119KB)
𝒅𝒚
Differential calculus 𝒅𝒙
For an introduction to differentiation:
• A brief refresher on basic differentiation, critical points and their nature, and with
applications to economics.
Introduction to calculus (pdf, 78KB)
• A more in-depth treatment to differentiation: rates of change, tangents and
derivatives, the product, quotient and chain rule, stationary points and optimisation
problems.
Introduction to differential calculus (pdf, 2.1MB)
For specific help on calculating derivatives using the rules of differentiation:
• Differentiating constants 𝑦 = 𝑘, polynomial functions 𝑦 = 𝑥 𝑛 , constant multiples 𝑦 =
𝑐𝑓(𝑥), addition and subtraction of functions 𝑦 = 𝑓(𝑥) ± 𝑔(𝑥), the product rule for
𝑢
𝑦 = 𝑢𝑣, and the quotient rule for 𝑦 = 𝑣 .
The rules of calculus (pdf, 89KB)
• The chain rule for composite functions 𝑦 = ℎ(𝑔(𝑥)), and its two formulations:
𝑑𝑦 𝑑𝑦 𝑑𝑢
𝑦 ′ = ℎ′ (𝑔(𝑥))𝑔′(𝑥) and = 𝑑𝑢 × 𝑑𝑥.
𝑑𝑥
Composite function rule (the chain rule) (pdf, 88KB)
• For derivatives of functions with exponentials 𝑒 𝑥 and logarithms ln 𝑥. Some of the
examples assumes knowledge of the chain rule.
Derivatives of exponential and logarithmic functions (pdf, 81KB)
• For derivatives of functions with sin 𝑥 , cos 𝑥 , tan 𝑥. Some of the examples assumes
knowledge of the product, quotient and chain rules.
Differentiation of trigonometry (pdf, 78KB)
For what derivatives can tell us about the shape of a graph:
𝑑𝑦
• The first derivative 𝑑𝑥 : increasing and decreasing functions, stationary points and their
nature, relative maximum and minimum.
The first derivative and stationary points (pdf, 99KB)
𝑑2 𝑦
• The second derivative 𝑑𝑥 2: concave up, concave down and points of inflection.
Second derivative and points of inflection (pdf, 95KB)
Integral calculus ∫ 𝒇(𝒙)𝒅𝒙
For an introduction to the indefinite integral ∫ 𝑓(𝑥)𝑑𝑥: anti-derivatives, calculating some
elementary anti-derivatives and reversing the Chain Rule.
Introduction to integration part 1: the anti-derivative (pdf, 191KB)
𝑏
For an introduction to the definite integral ∫𝑎 𝑓(𝑥)𝑑𝑥: limiting sums, the Fundamental Theorem
of Calculus, and finding areas under and between curves.
Introduction to integration part 2: the definite integral (pdf, 281KB)
Applications of calculus
For demonstrations of how to use the concepts and tools of differential calculus to sketch
graphs and curves of functions through several worked examples.
Curve sketching using calculus (pdf, 119KB)
