Lecture Notes on Differentiation

Introduction

• Topic: Differentiation
• Key focus: Understanding derivatives from various perspectives.
• Importance: Derivatives play a crucial role in science, engineering, economics, and more.

Overview of Upcoming Topics

• Main Topic: What is a derivative?
• Focus Areas:
  • Geometric interpretation of derivatives
  • Physical interpretation of derivatives
  • Importance of derivatives in measurements across different fields
  • How to differentiate any function

Geometric Interpretation of Derivatives

• Tangent Line Problem:
  • Goal: Find the tangent line to a graph at a specific point (x0, y0).
  • Visual representation: Graph a function and identify the tangent line at point P.

Tangent Line Equation

• Equation form: y - y0 = M(x - x0)
  • Where M is the slope of the tangent line.
• Two Required Pieces of Information:
  1. The point P (x0, f(x0)) on the curve.
  2. The slope M, defined as the derivative at x0 (f'(x0)).

Definition of Derivative

• Derivative Definition:
  • The derivative f'(x0) is the slope of the tangent line to the graph of f at the point P.
• Geometric Approach:
  • The slope of the tangent line can be approached through secant lines.
  • As the second point (Q) approaches P, the slope of the secant line approaches the slope of the tangent line.

Secant Lines and Slope Calculation

• Slope of Secant Line:

  • Formula: Slope = (f(Q) - f(P)) / (Q - P) where Q gets closer to P.
  • Notation: Delta F = change in function value, Delta X = change in x.

• Limit Notation:

  • The slope of the tangent line is the limit of the slope of the secant line as Delta X approaches 0:

  f'(x0) = lim (Delta X -> 0) (f(x0 + Delta X) - f(x0)) / Delta X

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

• Using the Derivative Formula:

  • Start with: Delta F / Delta X = (f(x0 + Delta X) - f(x0)) / Delta X
  • Substitute: f(x) = 1/x
  • Apply the limit process and simplify to get:

  f'(x0) = -1 / x0^2

Graphical Verification

• Check the sign of the derivative: It indicates the slope is negative, confirming the tangent line's behavior.
• As x0 approaches infinity, the slope decreases in magnitude.

Word Problems and Calculus Applications

• Discussed finding areas of triangles formed by tangents to curves.
• Importance of identifying tangent lines to solve geometry problems.

Additional Notations for Derivatives

• Common notations:
  • f'(x) or dy/dx
  • Df/Dx or d/dx(F)
• Multiple notations can be used interchangeably, depending on context.

Example 2: Derivative of f(x) = x^n

• Objective: Find the derivative of f(x) = x^n.
• Using Binomial Theorem:
  • Expand and simplify the expression for Delta F / Delta X.
  • Result: f'(x) = n*x^(n-1).
• Application to Polynomials:
  • Derivative can be applied to polynomial functions directly.*

Conclusion

• Key Takeaways:
  • Understanding the geometric and practical significance of derivatives.
  • Ability to calculate derivatives for various functions, including polynomials.
• Future Topics: Further applications and more examples in calculus.