Lecture Notes: 18.01 - Differentiation (Unit 1)

Introduction

• MIT OpenCourseWare – free high-quality resources (ocw.mit.edu).
• Unit 1: Differentiation.

Topics Covered

• What is a Derivative?
  • Geometric Interpretation.
  • Physical Interpretation.
  • Importance in science, engineering, economics, etc.
• How to Differentiate Any Function.

Geometric Interpretation of Derivatives

• Problem: Finding the tangent line to a graph at a point (x0, y0).
  • Example: Graph a function and determine the tangent line visually and analytically.
• Equation of a Tangent Line:
  • General form: y - y0 = M(x - x0).
  • Slope (M) is key to defining the tangent line.
• Defining the Slope: Frime (Derivative) of x0:
  • Slope equals the limit of the secant lines as they converge on the tangent point.
  • **Mathematical Definition of Derivative:
    • (f'(x_0) = \lim_{{\Delta x \to 0}} \frac{{f(x_0 + \Delta x) - f(x_0)}}{{\Delta x}}).

Example Calculation: Derivative of f(x) = 1/x

• Plug into derivative formula: ( \frac{{1/(x_0 + \Delta x) - 1/x_0}}{{\Delta x}} ).
  • Common denominator, simplify expressions.
  • Result: (f'(x_0) = -\frac{1}{{x_0^2}}).

Applying Differentiation Practically

• Finding area of triangles enclosed by the axes and tangent lines to y = 1/x.
  • Intersection points (x, y intercepts) using symmetry and tangent line equations.
  • Final Formula: Area = 2.

Notations in Differentiation

• Newton's Notation: f'(x), represents deltay, derivative notation.
• Leibniz's Notation: dy/dx or d(f)/dx, alternative notations.
  • Sometimes written: d by dx (of y or f).

More Advanced Examples

Example 2: Derivative of Powers of x

• General Formula: Derivative with respect to x of x^n = nx^(n-1).
  • Delta F / Delta X approach with binomial theorem simplifications.
  • Final Result (Formula): d/dx (x^n) = nx^(n-1).

Applications to Polynomials

• Calculating derivatives for polynomial expressions:
  • Example: d/dx (x^3 + 5x^10) = 3x^2 + 50x^9.

Key Takeaways

• Calculation of derivatives for various functions.
• Practical application in geometric problems.
• Importance of understanding multiple variable interactions in calculus.
• Understanding different derivative notations and their applications.

Next Class Preview

• Further exploration of differentiation applications and techniques.