## Derivative Patterns

Read this section to learn about patterns of derivatives. Work through practice problems 1-8.

The derivative of a function is a new function , and we can calculate the derivative of this new function to get the derivative of the derivative of , denoted by and called the second derivative of . For example, if then and .

**Definitions:** The first derivative of is , the rate of change of .

The second derivative of is , the rate of change of . The third derivative of is , the rate of change of ".

**Practice 8:** Find , and for , and

If represents the position of a particle at time , then will represent the velocity (rate of change of the position) of the particle and will represent the acceleration (the rate of change of the velocity) of the particle.

**Example 5:** The height (feet) of a particle at time seconds is . Find the height, velocity and acceleration of the particle when , and seconds.

Solution: so feet, feet, and feet

The velocity is so , and . At each of these times the velocity is positive and the particle is moving upward, increasing in height.

We will examine the geometric meaning of the second derivative later.