Derivative of a Function

Chapter 3.1

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Definition of the Derivative

• In the previous chapter, we defined the slope of the tangent line to a curve at a point as

• When this limit exists, it is called the derivative of at

• In this chapter we study the derivative as a function derived from by considering the limit at each point in the domain of

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Definition of the Derivative

DEFINITION:The derivative of the function with respect to the variable is the function (“f prime”) whose value is

provided the limit exists.

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Definition of the Derivative

• It is important to remember that is a function

• The domain of this function may be smaller than that of

• If exists, we say that is differentiable at (or that has a derivative at )

• A function that is differentiable at every point of its domain is a differentiable function

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Example 1: Applying the Definition

Find the derivative of .

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Example 1: Applying the Definition

Find the derivative of .

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Example 1: Applying the Definition

• The derivative of is

• IMPORTANT: remember that the value of at any value of is the slope of the tangent line at !

• If we need to find the slope of the tangent line at, say, , then

• That is, the slope of the tangent line at is 12

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Example 1: Applying the Definition

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Alternate Definition

DEFINITION:The derivative of the function at the point is the limit

provided the limit exists.

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Example 2: Applying the Alternate Definition

Differentiate (i.e., find the derivative) using the alternate definition.

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Example 2: Applying the Alternate Definition

Differentiate (i.e., find the derivative) using the alternate definition.

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Example 2: Applying the Alternate Definition

Differentiate (i.e., find the derivative) using the alternate definition.The result, is the derivative at some point . We can now let be any number in the domain of at which the limit exists and write

Note that, although , is not defined.

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Notation

• The following notations can be used to represent a derivative• “y prime”

• “dy dx” or “the derivative of y with respect to x”

• “df dx” or “the derivative of f with respect to x”

• “d dx of f at x” or “the derivative of f at x”

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Example 3: Graphing from

Graph the derivative of the function whose graph is shown in the next slide. Discuss the behavior of in terms of the signs and values of .

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Example 3: Graphing from

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Example 3: Graphing from

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Example 4: Graphing from

Sketch the graph of a function that has the following properties:

i.

ii. The graph of , the derivative of , is as shown on the next slide

iii. is continuous for all

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Example 4: Graphing from

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Example 4: Graphing from

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Example 5: Downhill Skier (Prob. 29)

The table below gives the approximate distance traveled by a downhill skier after seconds for . Sketch a graph of the derivative then answer the following questions.

a) What does the derivative represent?

b) In what units would the derivative be measured?

c) Guess an equation of the derivative by considering its graph.

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Example 5: Downhill Skier (Prob. 29)

Time (seconds) Distance (feet)

0 0

1 3.3

2 13.3

3 29.9

4 53.2

5 83.2

6 119.8

7 163.0

8 212.9

9 269.5

10 332.7

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Example 5: Downhill Skier (Prob. 29)

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Example 5: Downhill Skier (Prob. 29)

The table below gives the approximate distance traveled by a downhill skier after seconds for . Sketch a graph of the derivative then answer the following questions.

a) What does the derivative represent? Speed of skier

b) In what units would the derivative be measured? Feet per second

c) Guess an equation of the derivative by considering its graph.

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Example 5: Downhill Skier (Prob. 29)

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One-Sided Derivatives

DEFINITION:A function is differentiable on a closed interval if it has a derivative at every interior point of the interval, and if the limits

exist at the endpoints.

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One-Sided Derivatives

• At the left endpoint of a closed interval, is positive and approaches zero from the right (i.e., from values that are greater than zero)

• At the right endpoint, is negative and approaches zero from the left (i.e., from values that are less than zero)

• We may determine a closed interval (and, thus, right- and left-hand derivatives) at any two distinct points of a function’s domain

• As you saw in the previous chapter, a function has a (2-sided) derivative at a point if and only if the right-hand and left-hand limits exist at that point

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Example 6: One-Sided Derivatives Can Differ at a PointShow that the following function has left-hand and right-hand derivatives at , but no derivative there.

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Example 6: One-Sided Derivatives Can Differ at a PointShow that the following function has left-hand and right-hand derivatives at , but no derivative there.

For the left-hand limit (for )

For the right-had limit (for )

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Example 6: One-Sided Derivatives Can Differ at a PointShow that the following function has left-hand and right-hand derivatives at , but no derivative there.

Since the left- and right-had limits are different at , we say that the function is not differentiable at

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Exercise 3.1