Introduction to Differentiation

Jul 8, 2024

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