Introduction

This second lecture in the calculus series introduces derivatives and their computation. The professor explains calculus deals with pairs of functions: distance/speed and height/slope. The core problem presented is recovering the second function (speed/slope) when only the first function (distance/height) is known.

Central Concept

The lecture distinguishes between two types of measurements: total accumulation (distance, height) recorded over time versus instantaneous readings (speed, slope) at specific moments. The notation dy/dx represents the derivative - the ratio of vertical change to horizontal change, giving slope.

Three Great Functions

The professor identifies three fundamental calculus functions and their derivatives:

These functions combine through multiplication, division, and composition (chains) to create more complex functions, requiring product, quotient, and chain rules.

Detailed Example: y = x²

The lecture works through computing the derivative of x² step by step. Starting at x=0, the professor shows the slope is 0 (flat at the bottom). For any point x, the process involves:

1. Taking a small increment delta x
2. Finding the height change: (x + delta x)² - x²
3. Simplifying to get delta y/delta x = 2x + delta x
4. Taking the limit as delta x approaches 0
5. Result: dy/dx = 2x

Key Observations

Average slopes (like traveling from x=1 to x=2 on the parabola, giving average slope 3) differ from instantaneous slopes. Calculus focuses on instantaneous values by taking limits of increasingly smaller intervals.

Maximum and minimum points are identified where the slope equals zero - a crucial application for optimization problems.

Sine and Cosine Relationship

The lecture concludes by examining how the sine function's slope behavior matches the cosine function. When sine x reaches its maximum at π/2, the slope (cosine) equals zero. The slope is initially positive but decreasing, becomes zero at the peak, then turns negative as the function descends.

Conclusion

The lecture establishes the fundamental process of differential calculus: moving from a known function to finding its rate of change at every point. This foundation enables understanding of optimization, rates of change, and the behavior of functions in science, engineering, and economics.