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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
:
Plug f(x) into derivative formula
Algebraic simplifications
Use binomial formula for simplification
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
:
Find x-intercept by setting y = 0 in the tangent line equation
Find y-intercept by symmetry (or setting x = 0)
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
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