3.1 Derivative of a Function

What you’ll learn•Definition of a derivative•Notation•Relationships between the graphs of f and f’•Graphing the derivative from data•One-sided derivatives

Why? The derivative gives the value of the slope of the tangent line to a curve at a point – rate of change!

0

limh

f a h f a

h

is called the derivative of at .f a

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

What is a Derivative?

In chapter 2 we defined the slope of the curve y=f(x) at the point where x = a to be

When it exists, this limit is called “the derivative of f at a.” Now we will look at the derivative as a function derived from f by considering the limit (slope) at each point of the domain of f.

The derivative of the function f with respect to the variable x is the function f’ whose value at x is

Provided the limit exists.

h

afhaf

h

)()(

0

lim

h

xfhxf

hxf

)()(

0

lim)('

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

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.

Use the definition of derivative to find the derivative of f(x) = x2.

h

xfhxf

hxf

)()(

0

lim)('

Derivative at a Point

The derivative of the function f at the point

x = a is the limit

Provided the limit exists.

ax

afxf

axxf

)()(lim)('

Differentiate f(x) = x

ax

afxf

axxf

)()(lim)('

NotationThere are lots of ways to denote the derivative of a function y = f(x).

f’(x) the derivative of f

the derivative of f with

y’ y prime respect to x.

the derivative of y the derivative of f at x

with respect to x. dx

dy

dx

df

)(xfdx

d

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

Relationship between the graphs of f and f’

When we have a formula for f(x), we can derive a formula for f’(x) using methods like examples 1 & 2. Because we think of the derivative at a point in graphical terms as slope, we can get a good idea of what the graph of the function f’ looks like by estimating the slopes at various points along the graph of f.

How?

Draw the axis, marking the horizontal axis as x-units and the vertical axis as slope units.

Estimate the slope of the graph of f(x) at various points, plotting the slope values using the new axis.

Connect the plotted points with a smooth curve.

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.

Comparison: f and f’

Graph of f

• Increasing

• Decreasing

• Maximum or minimum value

(when slope = 0)

Graph of f’

• Positive (above x axis)• Negative (below x axis)• Zero

You try: Graphing f from f’

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

i) f(0) = 0

ii) The graph of f’, the derivative of f, is shown in F 3.4

iii) F is continuous for all x.

If f ’ is constant, f will be linear with that slope.

If f ’ is discontinuous, f is not differentiable at that point.

Anywhere f ’ is positive, f is increasing.

Anywhere f ’ is negative, f is decreasing.

Anywhere f ’=0, f has a maximum or minimum value.

Homework

p105 1-19(odds), 14, 16

Opener

p108 Exercises 36-41

No Calculator!

Graphing the Derivative from Data

You are given a table of data.

Create a second table, finding the slopes between each data points.

Plot f’(x) by plotting your points

(midpoint between 2 data points, slope at midpoint)

])/[],([ 21 xyxx

Practice

Data Table

0 05 0.02710 0.11715 0.25320 0.41125 0.56930 0.706

Estimates of slopes at midpoints

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 right hand derivative at a and the left hand derivative at b exist.

As with limits, if the functions right-hand and left-hand derivatives exist and are equal at a point, then the function is differentiable at that point.

One-Sided Derivatives can Differ at a Point.

Show that the following function is not differentiable at x = 0.

0,2

0,{

2

xx

xxy

Homework

page 105

Exercises 21-31 (odds)

Graph derivatives from functions

worksheet