Lecture Notes: Introduction to Derivatives

Overview

What is a Derivative?
- Geometric interpretation
- Physical interpretation
- Importance in various fields (science, engineering, economics, etc.)
- Ability to differentiate any function
Geometric Interpretation of Derivatives
- Problem: Finding the tangent line at point (x0, y0)
- Visualization of the tangent line
- Secant lines approaching tangent line
Tangent Line Equation
- Equation: ( y - y_0 = m(x - x_0) )
- Need to find point P (y = f(x0)) and slope m (derivative ( f' ))
- Definition: ( f'(x_0) ) is the slope of the tangent line to ( y = f(x) ) at point P.
Finding the Slope
- Slope = limit of secant line slope as point Q approaches P
- Formula: ( m = \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} )
Example Calculation
- Function: ( f(x) = \frac{1}{x} )
- Derivative formulation:
  - Calculate ( \Delta f ) and ( \Delta x )
  - Simplification and limit to find derivative
  - Result: ( f'(x_0) = -\frac{1}{x_0^2} )
Word Problem
- Problem: Finding areas of triangles formed by axes and tangent to ( y = \frac{1}{x} )
- Use of tangent line slope to find area
- Area formula: Area = ( \frac{1}{2} \times \text{base} \times ext{height} )
- Result: Area is constant (2) for the function ( y = \frac{1}{x} )

Different notations include:
- ( f' )
- ( \frac{df}{dx} )
- ( \frac{dy}{dx} )
- ( D_y f ) or ( D_x f )

Example 2: Derivative of ( f(x) = x^n )
Derivation using limit:
- Use binomial theorem for ( (x + \Delta x)^n )
- Result: ( \frac{d}{dx}(x^n) = nx^{n-1} )
Extension to polynomials:
- Example: ( \frac{d}{dx}(x^3 + 5x^{10}) = 3x^2 + 50x^9 )