Two problems that mathematicians were facing led to the discovery of differential calculus:

(1) how to define the instantaneous velocity, acceleration, and rates of change;

(2) how to construct the tangent line to a given curve at a point.

In this chapter, we will start with these two problems to develop the concept of a derivative and show what it means geometrically. Then we will learn how to find the derivatives of various functions. In more detail, we will cover:

- The Derivative Concept
- Geometric Interpretation of the Derivative as a Slope
- Graphing the Derivative
- Differentiability and Continuity of Functions
- One-Sided Derivatives
- When a Function Is Not Differentiable at a Point
- Differentiation Rules
- Derivatives of The Trigonometric Functions
- Higher Derivatives
- * More About the Leibniz Notation for Higher Derivatives
- Implicit Differentiation
- Derivatives of Inverse Functions
- Derivatives of the Inverse Trigonometric Functions
- Derivatives of Logarithmic Functions
- Derivatives of Exponential Functions
- Hyperbolic Functions and Their Derivatives
- Linear Approximations
- Differentials

The applications of derivatives will be covered in the next chapter.