Lecture: Introduction to Differentiation

Course Information

• Course: 18.01 (Calculus)
• Institute: MIT OpenCourseWare
• License: Creative Commons License
• URL: ocw.mit.edu

Unit 1: Differentiation

Overview

• Main Topic: What is a Derivative?
• Perspectives:
  • Geometric Interpretation
  • Physical Interpretation
  • Importance in measurements (science, engineering, economics, etc.)
• Objective: Explain how to differentiate any function
  • Example: Differentiate complex functions like e^x * arctan(x)

Geometric Interpretation of Derivatives

• Problem: Finding the tangent line to a graph of a function at a point (x0, y0)
  • Graph the function
  • Identify a point P at (x0, y0)
  • Draw the tangent line at that point
• Key Concepts:
  • Equation of a line through a point: y - y0 = m(x - x0)
  • Slope (m) is the derivative at point x0 (notation: f’(x0))
  • Definition: f’(x0) = slope of the tangent line to y = f(x) at point P
• Analytical Approach:
  • Use secant lines (line passing through two points on a curve) to approach the slope of the tangent line
  • Tangent line slope = limit of slopes of secant lines as Q approaches P

Algebraic Formulation

• Slope of Secant Line:
  • Change in X: Δx
  • Change in Y (f): Δf
  • Slope of Secant Line: Δf/Δx
  • Tangent Line Slope: limit as Δx -> 0 of Δf/Δx
• Concrete Formula:
  • Δf = f(x0 + Δx) - f(x0)
  • f’(x0) = limit as Δx -> 0 of [f(x0 + Δx) - f(x0)] / Δx

Example Calculation: f(x) = 1/x

• Objective: Compute derivative of 1/x
• Steps:
  1. Plug f(x) into derivative formula
  2. Algebraic simplifications
  3. Use binomial formula for simplification
  4. Take the limit as Δx -> 0
• Result: f’(x0) = -1/(x0^2)
• Geometric Interpretation: Slope of tangent line is negative, becomes less steep as x increases

Word Problem Example

• Problem: Find areas of triangles enclosed by the axes and the tangent to y = 1/x
• Solution:
  1. Find x-intercept by setting y = 0 in the tangent line equation
  2. Find y-intercept by symmetry (or setting x = 0)
  3. Calculate area of the triangle: 1/2 * base * height
• Result: Area = 2 (regardless of x0)
  • Demonstrates deeper understanding of the geometric visualization and implication of calculus concepts

Notations and Extensions

• Various Notations for Derivatives:
  • Newton’s Notation: f’(x)
  • Leibniz's Notations: df/dx, dy/dx, d/dx[f], d/dx[y]
• Extension to Polynomials:
  • General formula for derivative of x^n: d/dx [x^n] = nx^(n-1)
  • Example: d/dx [x^3 + 5x^10] = 3x^2 + 50x^9

Binomial Theorem Utilization

• Binomial Theorem: Used to simplify complex terms involving powers
• Purpose: Simplifies algebra before taking limit

Key Takeaways

• Understanding derivative as the slope of tangent line
• Differentiating complex functions using fundamental limits and algebra
• Applying calculus in geometric contexts and beyond (practical implementations)
• Mastering notation and general rules for calculating derivatives