Definition of the Derivative

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Definition of the Derivative. Section 3.1a. Answers to the “Do Now” – Quick Review, p.101. 1. 7. 2. 8. 3. 9. No, the one-sided limits at x = 1 are different. 4. 5. Slope:. 10. No, f is discontinuous at x = 1 because. 6. DNE.

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### Definition of the Derivative

Section 3.1a

Answers to the “Do Now” – Quick Review, p.101

1.

7.

2.

8.

3.

9. No, the one-sided limits

at x = 1 are different

4.

5. Slope:

10. No, f is discontinuous

at x = 1 because

6.

DNE

on the graph of the function f(x) = 1/x :

This study of the rates of change of functions is called

differential calculus, and the formula was

our first look at a derivative…

Definition: Derivative

The derivative of the function with respect to the variable x is

the function whose value at x is

provided the limit exists.

Note: The domain of may be smaller than that of .

If exists, we say that has a derivative (is

differentiable) at x. A function that is differentiable at every

point of its domain is a differentiable function.

Applying the Definition

Differentiate (that is, find the derivative of)

Now, let’s see what happens when we re-label our diagram

Slope of the secant line PQ:

And what limit should we take here to find the derivative???

Definition (Alternate): Derivative at a Point

The derivative of the function f at the point x = ais the limit

provided the limit exists.

If we use this alternate form to find the derivative of f at x = a,

we can find the general derivative of f by applying our answer

to an arbitrary x in the domain of f…………………observe……

Applying Both Definitions

Differentiate using both derivative definitions.

The original definition:

Applying Both Definitions

Differentiate using both derivative definitions.

The alternate definition:

Applying this formula to an arbitrary x > 0 in the domain of f

identifies the derivative as the function

Besides , the most common notations for the derivative

of a function :

Notation

Notes

“y prime”

Nice and brief, but does

not name the independent

variable.

“dy/dx” or “the derivative

of y with respect to x”

Names both variables and

uses d for derivative.

Besides , the most common notations for the derivative

of a function :

Notation

Notes

“df/dx” or “the derivative

of f with respect to x”

Emphasizes the function’s

name.

“d dxof f at x” or “the

derivative of f at x”

Emphasizes the idea that

differentiation is an

operation performed on f.

One-Sided Derivatives

A function y = f(x) is differentiable on a closed interval [a,b] if

it has a derivative at every interior point of the interval, and if the

limits

[the right-hand derivative at a]

[the left-hand derivative at b]

exist at the endpoints.

Note: In the right-hand derivative, h is positive and a + h

approaches a from the right. In the left-hand derivative, h is

negative and b + h approaches b from the left.

One-Sided Derivatives

Derivatives at endpoints are one-sided limits:

Slope =

Slope =

One-Sided Derivatives

Show that the following function has left-hand and right-hand

derivatives at x= 0, but no derivative there.

Graph the function:

The left-hand derivative:

The right-hand derivative:

Since these are not

equal, the function has

no derivative at x = 0!!!