## Derivative Patterns

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

If we apply the Product Rule to the product of a function with itself, a familiar pattern emerges.

**Practice 1:** What is the pattern here? What do you think the results will be for and ?

We could keep differentiating higher and higher powers of by writing them as products of lower powers of and using the Product Rule, but the Power Rule For Functions guarantees that the pattern we just saw for the small integer powers also works for all constant powers of functions.

**Power Rule For Functions:** If is any constant,

The Power Rule for Functions is a special case of a more general theorem, the Chain Rule, which we will examine in Section 2.4. The Power Rule For Functions will be proved after the Chain Rule.

**Example 1:** Use the Power Rule for Functions to find

Solution: (a) To match the pattern of the Power Rule for , let and .

(b) To match the pattern for , we can let and take . Then

(c) To match the pattern for , Let and . Then

**Practice 2:** Use the Power Rule for Functions to find

**Example 2:** Use calculus to show that the line tangent to the circle at the point has slope .

Solution: The top half of the circle is the graph of so

As a check, you can verify that the slope of the radial line through the center of the circle and the point has slope and is perpendicular to the tangent line which has a slope of .