Section 2.8Section 2.8

The The Derivative as a Derivative as a FunctionFunction

GoalsGoals View the derivative View the derivative f f ´(´(xx) as a ) as a functionfunction

of of xx Study graphs of Study graphs of f f ´(´(xx) and ) and f f((xx) together) together

Study Study differentiabilitydifferentiability and and continuitycontinuity Introduce Introduce higher-order derivativeshigher-order derivatives

IntroductionIntroduction

So far we have considered the So far we have considered the derivative of a function derivative of a function ff at a at a fixedfixed number number aa : :

Now we change our point of view Now we change our point of view and let the number and let the number aa varyvary::

0

limh

f a h f af a

h

0

limh

f x h f xf x

h

Introduction (cont’d)Introduction (cont’d)

Thus Thus f f ´(´(xx) becomes its own, new, ) becomes its own, new, function of function of xx , called the , called the derivative derivative of fof f . . This name reflects the fact that This name reflects the fact that f f ´ has ´ has

been “derived” from been “derived” from ff . .

Note that Note that f f ´(´(xx) is a ) is a limitlimit.. Thus Thus f f ´(´(xx) is defined only when this ) is defined only when this

limit exists.limit exists.

ExampleExample

At right is the At right is the graph of a function graph of a function ff . .

We want to use We want to use this graph to this graph to sketch the graph of sketch the graph of the derivative the derivative f f ´(´(xx) .) .

SolutionSolution

We can estimate f f ´(´(xx) ) at any x by drawing the tangent at the point (x, f(x))

and estimating its slope.

Thus, for x = 5 we draw the tangent at P in Fig. 2(a) (on the next slide), and estimate f f ´(5) ≈ 1.5 .´(5) ≈ 1.5 . Then we plot P P ´(5, 1.5)´(5, 1.5) on the graph of f f ´ ´ .

Repeating gives the graph in Fig. 2(b).

Solution (cont’d)Solution (cont’d)

Solution (cont’d)Solution (cont’d)

Remarks on the SolutionRemarks on the Solution

The tangents at A , B , and C are horizontal, so the derivative is 0 there, and the graph of f f ´́ crosses the x-axis at A A ´́,

B  ´́, and CC´́, directly beneath A, B, and C.

Between… A and B , f f ´(´(xx) ) is positive; B and C , f f ´(´(xx)) is negative.

ExampleExample

For the function For the function ff((xx) = ) = xx33 – – xx , , Find a formula for Find a formula for f f ´(´(xx)) Compare the graphs of Compare the graphs of ff and and f f ´́

SolutionSolution On the… On the…a)a) next slide, we show that next slide, we show that f f ´(´(xx) = 3) = 3xx22 – –

1 ;1 ;

b)b) following slide, we give the graphs of following slide, we give the graphs of ff and and f f ´ side-by-side:´ side-by-side:

Solution (cont’d)Solution (cont’d)

Solution (cont’d)Solution (cont’d)

Notice that Notice that f f ´(´(xx) is…) is… zero when zero when ff has horizontal tangents, has horizontal tangents,

andand positive when the tangents have positive when the tangents have

positive slope:positive slope:

ExampleExample

Find Find f f ´(´(xx) if) if

SolutionSolution We use the definition as We use the definition as follows:follows:

1.

2x

f xx

Solution (cont’d)Solution (cont’d)

Other NotationsOther Notations

Here are common alternative notations Here are common alternative notations for the derivative:for the derivative:

The symbols The symbols DD and and dd//dxdx are called are called differentiation operatorsdifferentiation operators because they because they indicate the operation of indicate the operation of differentiationdifferentiation, the process of , the process of calculating a derivative.calculating a derivative.

Other Notations (cont’d)Other Notations (cont’d) The Leibniz symbol The Leibniz symbol dydy//dxdx is not an is not an

actualactual ratio, but rather a ratio, but rather a synonymsynonym for for f f ´(´(xx) .) .

We can write the definition of We can write the definition of derivative as:derivative as:

Also we can indicate the value Also we can indicate the value f f ´(´(aa) ) of a derivative of a derivative dydy//dxdx as as

0limx

dy ydx x

or x a x a

dy dydx dx

DifferentiabilityDifferentiability

We begin with this definition:We begin with this definition:

This definition captures the fact that This definition captures the fact that some functions have derivatives only some functions have derivatives only at some values of at some values of xx , not all. , not all.

ExampleExample

Where is the function Where is the function ff((xx) = ) = ||xx|| differentiable?differentiable?

SolutionSolution If If xx > 0 , then… > 0 , then… ||xx|| = = xx and we can choose and we can choose hh small small

enough that enough that xx + + hh > 0 , so that > 0 , so that ||x + hx + h|| = = x + hx + h

ThereforeTherefore

0 0 0lim lim lim 1h h h

x h x x h x hf x

h h h

Solution (cont’d)Solution (cont’d) This means that This means that ff is differentiable is differentiable

for any for any xx > 0 . > 0 . A similar argument shows that A similar argument shows that ff is is

differentiable for any differentiable for any xx < 0 , as well. < 0 , as well.

However for However for xx = 0 we have to = 0 we have to considerconsider

Solution (cont’d)Solution (cont’d)

We compute the left and right limits We compute the left and right limits separately:separately:

Since these differ, Since these differ, f f ´(0) does not ´(0) does not exist.exist.

Thus Thus ff is differentiable at is differentiable at all all xx ≠ 0 ≠ 0 . .

Solution (cont’d)Solution (cont’d)

We can give a formula for We can give a formula for f f ´(´(xx) :) :

Also, on the next slide we graph Also, on the next slide we graph ff and and f f ´ side-by-side:´ side-by-side:

Solution (cont’d)Solution (cont’d)

Differentiability and Differentiability and ContinuityContinuity

We can show that We can show that if if ff is is differentiable at differentiable at aa , then , then ff is is continuous at continuous at aa . .

However, as our preceding example However, as our preceding example shows, shows, the converse is falsethe converse is false::

The function The function ff((xx) = ) = ||xx|| isis continuous everywhere, but continuous everywhere, but is is notnot differentiable at differentiable at xx = 0 . = 0 .

Failure of Failure of DifferentiabilityDifferentiability

A function can A function can failfail to be differentiable to be differentiable at at xx = = aa in in threethree different ways: different ways:

The graph of The graph of ff can have a can have a cornercorner at at xx = = aa……

……as does the graph of as does the graph of ff((xx) = ) = ||xx|| ; ;

ff can be can be discontinuousdiscontinuous at at xx = = aa ; ; The graph of The graph of ff can have a can have a vertical vertical

tangent linetangent line at at xx = = aa . . This means that This means that ff is continuous at is continuous at aa but but ||f f 

´́((xx))|| has an infinite limit as has an infinite limit as x x aa . .

We illustrate each of these possibilities:We illustrate each of these possibilities:

Corner at Corner at xx = = aa

Discontinuity at Discontinuity at xx = = aa

Vertical Tangent at Vertical Tangent at xx = = aa

More on DifferentiabilityMore on Differentiability

The next slides illustrate another way The next slides illustrate another way of looking at differentiability.of looking at differentiability.

We zoom in toward the point (We zoom in toward the point (aa, , ff((aa)) :)) : If If ff isis differentiable at differentiable at xx = = aa , then the , then the

graphgraph straightens outstraightens out and and appears more and more like a appears more and more like a lineline..

If If ff is is notnot differentiable at differentiable at xx = = aa , then , then nono amount of zooming makes the graph linear.amount of zooming makes the graph linear.

ff Is Differentiable At Is Differentiable At aa

ff Is Not Differentiable At Is Not Differentiable At aa

The Second DerivativeThe Second Derivative If If ff is a differentiable function, is a differentiable function,

then…then… its derivative its derivative f f ´ is also a function, so´ is also a function, so f f ´ may have a derivative of its ´ may have a derivative of its ownown, ,

denoted by (denoted by (f f ´)´ = ´)´ = f f  , and called the , and called the second derivativesecond derivative of of ff . .

In Leibniz notation the second In Leibniz notation the second derivative of derivative of yy = = ff((xx) is written) is written2

2

dy d yddx dx dx

ExampleExample

If If ff((xx) = ) = xx33 – – xx , find and interpret , find and interpret f f ((xx) .) .

SolutionSolution We found earlier that the first We found earlier that the first derivativederivative

f f ´(´(xx) = 3) = 3xx22 – 1 . – 1 . On the next slide we use the limit On the next slide we use the limit

definition of the derivative to show thatdefinition of the derivative to show that

f f ((xx) = 6) = 6xx : :

Solution (cont’d)Solution (cont’d)

Solution (cont’d)Solution (cont’d) On the next slide are the graphs of On the next slide are the graphs of ff , , f f ´ , ´ ,

and and f f  . . We can interpret We can interpret f f ((xx) as the slope of the ) as the slope of the

curve curve yy = = f f ´(´(xx) at the point () at the point (xx , , f f ´(´(xx)) .)) . That is, That is, f f ((xx) is the ) is the rate of changerate of change of the of the slopeslope

of the original curve of the original curve yy = = ff((xx) .) .

Notice in Fig. 11 thatNotice in Fig. 11 that f f ((xx) < 0 when ) < 0 when yy = = f f ´(´(xx) has a negative ) has a negative

slope;slope; f f ((xx) > 0 when ) > 0 when yy = = f f ´(´(xx) has a positive slope.) has a positive slope.

Solution (cont’d)Solution (cont’d)

AccelerationAcceleration If If ss = = ss((tt) is the position function of a ) is the position function of a

object moving in a straight line, then…object moving in a straight line, then… its its firstfirst derivative gives the derivative gives the velocityvelocity vv((tt) )

of the object:of the object:

The The accelerationacceleration aa((tt) of the object is the ) of the object is the derivative of the velocity functionderivative of the velocity function, that is, , that is, the the secondsecond derivative of the position derivative of the position function:function:

dsv t s t

dt

2

2, or in Leibniz notation, dv d s

a t v t s t adt dt

ExampleExample

A car starts from rest and the graph of A car starts from rest and the graph of its position function in shown on the its position function in shown on the next slide.next slide. Here Here ss is measured in feet and is measured in feet and tt in in

seconds.seconds.

Use this to graph the Use this to graph the velocityvelocity and and accelerationacceleration of the car. of the car.

What is the acceleration at What is the acceleration at tt = 2 = 2 seconds?seconds?

Position Function of a Position Function of a CarCar

SolutionSolution By measuring the slope of the graph ofBy measuring the slope of the graph of

ss = = ff((tt) at ) at tt = 0, 1, 2, 3, 4, and 5, we plot = 0, 1, 2, 3, 4, and 5, we plot the velocity function the velocity function vv = = f f ´(´(tt) (next ) (next slide).slide).

The acceleration when The acceleration when tt = 2 is = 2 is aa = = f f (2)(2)…… ……the slope of the tangent line to the graph of the slope of the tangent line to the graph of

f f ´ when ´ when tt = 2 . = 2 .

The slope of this tangent line isThe slope of this tangent line is

Velocity FunctionVelocity Function

Acceleration FunctionAcceleration Function In a similar way we can graph In a similar way we can graph aa((tt) :) :

Third DerivativeThird Derivative

The The third derivativethird derivative f f  is the is the derivative of the second derivative: derivative of the second derivative: f f  = ( = (f f )) . .

If If yy = = ff((xx) , then alternative ) , then alternative notations for the third derivative arenotations for the third derivative are

Higher-Order DerivativesHigher-Order Derivatives The process can be continued:The process can be continued:

The fourth derivative The fourth derivative f f  is usually is usually denoted by denoted by ff(4)(4) . .

In general, the In general, the nnth derivative of th derivative of ff is… is… denoted by denoted by ff((nn)) and and obtained from obtained from ff by differentiating by differentiating nn times. times.

If If yy = = ff((xx) , then we write) , then we write

ExampleExample

If If ff((xx) = ) = xx33 – 6 – 6xx , find , find f f ((xx) and ) and ff(4)(4)((xx) .) . SolutionSolution Earlier we found that Earlier we found that f f ((xx) = 6) = 6x x

.. The graph of The graph of yy = 6 = 6xx is a line with slope 6 ; is a line with slope 6 ; Since the derivative Since the derivative f f ((xx) is the slope of ) is the slope of f f 

((xx) , we have) , we have

f f ((xx) = 6 for all values of ) = 6 for all values of xx . . Therefore, for all values of Therefore, for all values of xx , ,

ff(4)(4)((xx) = 0) = 0

ReviewReview The derivative as a functionThe derivative as a function

The graph of The graph of f f  derived from the graph derived from the graph of of ff

Finding formulas for Finding formulas for f f ´(´(xx))

DifferentiabilityDifferentiability DefinitionDefinition Differentiability implies continuity…Differentiability implies continuity…

……but not converselybut not conversely

Higher-order derivativesHigher-order derivatives NotationNotation