3.2 DerivativesGreat Sand Dunes National Monument, Colorado

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

We write: 0

limh

f x h f xf x

h

“The derivative of f with respect to x is …”

There are many ways to write the derivative of y f x

f x “f prime x” or “the derivative of f with respect to x”

y “y prime”

dy

dx“dee why dee ecks” or “the derivative of y with

respect to x”

df

dx“dee eff dee ecks” or “the derivative of f with

respect to x”

df x

dx“dee dee ecks uv eff uv ecks” or “the derivative

of f of x”( of of )d dx f x

dx does not mean d times x !

dy does not mean d times y !

dy

dx does not mean !dy dx

(except when it is convenient to think of it as division.)

df

dxdoes not mean !df dx

(except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.)

df x

dxdoes not mean times !

d

dx f x

In the future, all will become clear.

y f x

y f x

The derivative is the slope of the original function.

The derivative is defined at the end points of a function on a closed interval.

2 3y x

2 2

0

3 3limh

x h xy

h

2 2 2

0

2limh

x xh h xy

h

2y x

0lim 2h

y x h

0

2Given: ( ) 2 3 Find: 'f x x x f x

2 2

0

2 3 2 3' lim

h

x h x h x xf

h

' 2 2f x

0' lim 2 2

hf x h

0

2 2 2

0

2 2 2 3 2 3' lim

h

x xh h x h x xf

h

A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

To be differentiable, a function must be continuous and smooth.

Derivatives will fail to exist at:

corner cusp

vertical tangent discontinuity

f x x 2

3f x x

3f x x 1, 0

1, 0

xf x

x

Most of the functions we study in calculus will be differentiable.

If a function is differentiable , then it is continuous.

Note!! Just because a function is continuous does not mean that it is differentiable!!!

ExampleShow that the following function is continuous but

not differentiable at x = 1. Sketch the graph of f.2 2, 1

( )2, 1

x xf x

x x

Derivatives on the TI-89:

You must be able to calculate derivatives with the calculator and without.

Today you will be using your calculator, but be sure to do them by hand when called for.

Remember that half the test is no calculator.

3y xExample: Find at x = 2.dy

dx

d ( x ^ 3, x ) ENTER returns23x

This is the derivative symbol, which is .82nd

It is not a lower case letter “d”.

Use the up arrow key to highlight and press .23x ENTER

3 ^ 2 2x x ENTER returns 12

or use: ^ 3, 2d x x x ENTER

Warning:

The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable.

Examples: 1/ , 0d x x x returns

, 0d abs x x x returns 1

Graphing Derivatives

Graph: ln ,y d x x What does the graph look like?

This looks like:1

yx

Use your calculator to evaluate: ln ,d x x1

x

The derivative of is only defined for , even though the calculator graphs negative values of x.

ln x 0x