Calculus 1 Lecture 2.1: Introduction to the Derivative of a Function

Definition and Meaning of the Derivative00:00:00

The derivative is introduced as a tool to find the instantaneous rate of change, instantaneous velocity, and the slope of a curve at a single point, represented by the limit of the difference quotient.

Finding the Derivative Function00:03:07

The lecture explains how to move beyond calculating the derivative at a single fixed point to finding a general 'derivative function' (f'(x)), which allows for calculating slopes at any x-value.

Example: Derivative of a Quadratic Function00:05:33

A step-by-step demonstration shows how to calculate the derivative of f(x) = 2x² - 3 and use that derivative to determine the equation of a tangent line at a specific point.

Example: Derivative of a Cubic Function00:16:00

An example involving a more complex function, f(x) = 2x³ - x, demonstrates how to apply the difference quotient and limit definition to find the derivative function.

Constant Slopes and Square Root Functions00:27:17

The video discusses the shortcut for linear functions where the slope is constant, and covers a more complex case involving a square root function (f(x) = √x) and its tangent line.

Applications: Instantaneous Velocity00:40:50

The derivative is applied to a position function to derive an instantaneous velocity function, including solving a problem about an object falling from a building.

Differentiability and Continuity00:56:49

The concept of differentiability is explained, noting that functions are not differentiable at sharp points or where the slope is undefined (vertical). It establishes that differentiability implies continuity, but continuity does not guarantee differentiability.

Derivative Notation and Evaluation01:11:47

The video concludes by summarizing various notations for derivatives (f'(x), dy/dx, d/dx) and demonstrating how to evaluate these derivatives at specific points.