YS

W7-8_Differentiation

Fundamentals of Mathematics 1: Differentiation

Instructor: Erwan LamyInstitution: ESCP BUSINESS SCHOOLLocations: Berlin, London, Madrid, Paris, Turin, Warsaw

Objectives

  • Compute derivatives using the limit definition.

  • Apply basic differentiation rules.

  • Interpret the derivative as an instantaneous rate of change.

  • Apply the product and quotient rules for differentiation.

  • Use the chain rule for function composition.

  • Find higher-order derivatives.

Outline

  • Chapter 11.1: The Derivative

  • Basic Rules for Differentiation: (Chapters 11.2, 11.4, 11.5)

  • Product and Quotient Rule: (Chapter 11.4)

  • Chain Rule

  • Derivative as a Rate of Change: (Chapter 11.3)

  • Applications of Rate of Change to Economics: (Chapter 11.3)

  • Higher-Order Derivatives: (Chapter 12.7)

The Derivative

Definition

The tangent line to a curve at point P is a straight line that touches the curve only at point P without crossing it nearby.

Slope of the Curve

The slope of the curve at point P corresponds to the slope of the tangent line at that point. The derivative, denoted as f'(x), provides the slope of this tangent line.Mathematical Definition:f'(x) = lim(h→0) [(f( x + h) - f(x))/h]

Rate of Change

The derivative f'(x) indicates the rate of change of the function f(x):

  • Positive f'(x): Function is increasing.

  • Negative f'(x): Function is decreasing.

  • The magnitude of f'(x) indicates the speed of increase or decrease.

Methods

  • Analytical Method: Use of the limit definition to compute derivatives.

  • Graphical Interpretation: Visualization of slopes of tangent lines on graphs.

Example

For f(x) = 2x^2 + 2x + 3, the slope of the tangent line is given by:f'(x) = 4x + 2.At x = 1, f'(1) = 4(1) + 2 = 6.

Notations

Expressing Derivatives

Various notations for expressing the derivative:

  • Leibniz's notation: dy/dx

  • Lagrange's notation: f'(x)

  • Newton's notation: f''(x)

  • Euler's notation: f^{(n)}(x)

Existence of a Derivative (Chapter 11.1)

Conditions for Existence

A derivative exists if the limit of the function is the same from both sides:lim(h→0-) f(a+h) = lim(h→0+) f(a+h).Geometrically, this means the left-hand and right-hand slopes at point a must be equal.

Example

Consider f(x) = { sqrt(x) for 0 ≤ x ≤ 1, 1 for x > 1}

  • Slope on left of x=1: 0.5

  • Slope on right of x=1: -1

The derivative at x=1 is not defined due to differing slopes.

Methods

  • Analytical Verification: Verify limits from both sides for existence.

  • Graphical Analysis: Plot functions and observe slopes at critical points.

Basic Rules for Differentiation (Chapter 11.2)

Power Rule and Basic Rules

  • If f(x) = x^n, then f'(x) = nx^(n-1).

Additional Basic Rules Include:

  • For constants: f'(c) = 0

  • For sums/differences: (f(x) ± g(x))' = f'(x) ± g'(x).

Further Examples

If f(x) = 6x^6, then f'(x) = 36x^5.

Methods

  • Tabular Method: Create tables to organize and compute derivatives systematically.

  • Practice Problems: Engage with different types of polynomial functions for mastery.

Product and Quotient Rule (Chapter 11.4)

Product Rule

(f(x)g(x))' = f'(x)g(x) + f(x)g'(x).

Quotient Rule

(f(x)/g(x))' = [f'(x)g(x) - f(x)g'(x)]/(g(x))^2.

Detailed Example (Product Rule)

If f(x) = (2+x)(4+x), apply the product rule. Use detailed substitution in calculating f'(x).

Detailed Example (Quotient Rule)

If f(x) = g(x)/h(x), apply the quotient rule properly, ensuring to differentiate and substitute carefully.

Methods

  • Step-by-Step Breakdown: Follow structured steps when applying these rules.

  • Collaborative Exercises: Work in pairs to enhance understanding through teaching.

Chain Rule (Chapter 11.5)

Formula

For composite functions, (f(g(x)))' = f'(g(x))g'(x).

Detailed Example

For f(x) = (2-x^3 + 1)^17, calculate derivatives through both Lagrange's and Leibniz's notations.

Methods

  • Function Decomposition: Break down complex functions into simpler components.

  • Visualization: Use diagrams to illustrate composite functions and their derivatives.

Higher-Order Derivatives (Chapter 12.7)

Notation

  • First derivative: f'(x)

  • Second derivative: f''(x)

  • Higher derivatives indicated similarly: f^{(n)}(x)

Example

For function f(x) = 6x^3 - 12x^2 + 6, compute higher-order derivatives. The third derivative results in zero, indicating no increase in order beyond that point.

Methods

  • Iterative Process: Gradually compute derivatives to obtain higher orders systematically.

  • Graphical Representation: Visualize the change in function behavior through higher derivatives.

Applications of Rate of Change in Economics (Chapter 11.3)

Total-Cost and Marginal Cost Functions

Marginal cost is calculated as the derivative of the total cost function.

Examples of Marginal Behavior

Understand how marginal propensity to consume and save are derived from respective functions.

Methods

  • Real-World Data Analysis: Apply differentiation to actual cost functions for better understanding.

  • Case Studies: Discuss economic theories and their mathematical underpinnings to link concepts with practical applications.

Exercise Instructions

Formulate varied exercises based on the established derivative principles and economic applications outlined.