Derivatives are the key to understanding how things change. They give us the slope of a curve at any point, telling us how fast something is increasing or decreasing. This concept is crucial for analyzing motion, rates of change, and optimization problems.

Velocity, a real-world application of derivatives, shows how an object's position changes over time. By finding the derivative of a position function, we can determine an object's speed and direction at any moment, bridging the gap between mathematical theory and practical applications.

The Derivative and Tangent Lines

Tangent line concept

• Straight line that touches a curve at a single point without crossing it (point of tangency)
• Represents the instantaneous direction of the curve at the point of tangency
• Slope of the tangent line indicates the rate of change of the curve at that point
  • Positive slope: curve is increasing
  • Negative slope: curve is decreasing
  • Zero slope: curve has a horizontal tangent line
• Tangent line is closely related to the secant line, which intersects the curve at two points

Tangent slope calculation

• Use the limit definition of the derivative to find the slope of a tangent line
  • $m_{tangent} = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$, where $x_0$ is the x-coordinate of the point of tangency
• Evaluate the limit of the difference quotient as h approaches zero
  • Difference quotient: average rate of change of the function over the interval $[x_0, x_0 + h]$
  • As h approaches zero, the interval becomes smaller, and the average rate of change approaches the instantaneous rate of change

Derivative as limit

• Derivative of a function $f(x)$ at a point $x_0$ is defined as the limit of the difference quotient as h approaches zero
  • $f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$
• Derivative function gives the instantaneous rate of change of the original function at any point
  • Denoted as $f'(x)$ or using Leibniz notation: $\frac{d}{dx}f(x)$

Derivative at specific point

• To find the derivative at a specific point, evaluate the limit of the difference quotient at that point
  1. Substitute the x-coordinate of the point into the difference quotient and simplify
  2. Evaluate the limit as h approaches zero
• Resulting value is the slope of the tangent line and the instantaneous rate of change at that point

Continuity and Differentiability

• Continuity is a prerequisite for differentiability
• A function is differentiable at a point if it is continuous at that point and its derivative exists
• Differentiability implies that the function has a well-defined tangent line at the point
• Newton's method for finding roots of equations relies on the differentiability of functions

Derivatives and Velocity

Velocity as rate of change

• Velocity is the rate of change of position with respect to time
• In the context of derivatives, velocity is the derivative of the position function
• Derivative of the position function gives the instantaneous velocity at any point in time
  • Positive velocity: object is moving in the positive direction
  • Negative velocity: object is moving in the negative direction
  • Zero velocity: object is momentarily at rest

Average vs instantaneous velocity

• Average velocity: total displacement divided by the total time elapsed
  • $v_{avg} = \frac{\Delta x}{\Delta t}$, where $\Delta x$ is the change in position and $\Delta t$ is the change in time
• Instantaneous velocity: velocity at a specific instant in time
  • Limit of the average velocity as the time interval approaches zero
  • $v_{inst} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$, where $\frac{dx}{dt}$ is the derivative of the position function

Derivative estimation from data

• Derivatives can be estimated using numerical data from tables or graphs
• Estimating derivatives from a table:
  1. Find the average rate of change between the point and a nearby point
  2. $\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two nearby points
  3. Choose points closer to the point of interest for a better approximation
• Estimating derivatives from a graph:
  1. Draw a tangent line to the curve at the point of interest
  2. Estimate the slope of the tangent line by calculating the rise over run using nearby points on the tangent line