Lesson 7: The Derivative as a Function

Section 2.2The Derivative as a Function

V63.0121.002.2010Su, Calculus I

New York University

May 24, 2010

Announcements

I Homework 1 due TuesdayI Quiz 2 Thursday in class on Sections 1.5–2.5

. . . . . .

. . . . . .

Announcements

I Homework 1 due TuesdayI Quiz 2 Thursday in class

on Sections 1.5–2.5

V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 2 / 28

. . . . . .

Objectives

I Given a function f, use thedefinition of the derivativeto find the derivativefunction f’.

I Given a function, find itssecond derivative.

I Given the graph of afunction, sketch the graphof its derivative.


Derivative. . . . . .

. . . . . .

Recall: the derivative

DefinitionLet f be a function and a a point in the domain of f. If the limit

f′(a) = limh→0

f(a+ h)− f(a)h

= limx→a

f(x)− f(a)x− a

exists, the function is said to be differentiable at a and f′(a) is thederivative of f at a.The derivative …

I …measures the slope of the line through (a, f(a)) tangent to thecurve y = f(x);

I …represents the instantaneous rate of change of f at aI …produces the best possible linear approximation to f near a.


. . . . . .

Derivative of the reciprocal function

Example

Suppose f(x) =1x. Use the

definition of the derivative tofind f′(2).

Solution

f′(2) = limx→2

1/x− 1/2x− 2

= limx→2

2− x2x(x− 2)

= limx→2

−12x

= −14

. .x

.x

.


. . . . . .


Example



Solution

f′(2) = limx→2

1/x− 1/2x− 2

= limx→2

2− x2x(x− 2)

= limx→2

−12x

= −14

. .x

.x

.


. . . . . .

Outline

What does f tell you about f′?

How can a function fail to be differentiable?

Other notations

The second derivative


. . . . . .


I If f is a function, we can compute the derivative f′(x) at each pointx where f is differentiable, and come up with another function, thederivative function.

I What can we say about this function f′?

I If f is decreasing on an interval, f′ is negative (technically,nonpositive) on that interval

I If f is increasing on an interval, f′ is positive (technically,nonnegative) on that interval


. . . . . .



I What can we say about this function f′?I If f is decreasing on an interval, f′ is negative (technically,

nonpositive) on that interval

I If f is increasing on an interval, f′ is positive (technically,nonnegative) on that interval


. . . . . .


Example



Solution

f′(2) = limx→2

1/x− 1/2x− 2

= limx→2

2− x2x(x− 2)

= limx→2

−12x

= −14

. .x

.x

.


. . . . . .



I What can we say about this function f′?I If f is decreasing on an interval, f′ is negative (technically,

nonpositive) on that intervalI If f is increasing on an interval, f′ is positive (technically,

nonnegative) on that interval


. . . . . .

Graphically and numerically

. .x

.y

..2

..4 .

.

..3

..9

.

..2.5

..6.25

.

..2.1

..4.41 .

..2.01

..4.0401

.

..1

..1

.

..1.5

..2.25

.

..1.9

..3.61.

..1.99

..3.9601

x m =x2 − 22

x− 23 52.5 4.52.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3


. . . . . .


FactIf f is decreasing on (a,b), then f′ ≤ 0 on (a,b).

Proof.If f is decreasing on (a,b), and ∆x > 0, then

f(x+∆x) < f(x) =⇒ f(x+∆x)− f(x)∆x

< 0

But if ∆x < 0, then x+∆x < x, and

f(x+∆x) > f(x) =⇒ f(x+∆x)− f(x)∆x

< 0

still! Either way,

f(x+∆x)− f(x)∆x

< 0 =⇒ f′(x) = lim∆x→0


≤ 0


. . . . . .




f(x+∆x) < f(x) =⇒ f(x+∆x)− f(x)∆x

< 0


f(x+∆x) > f(x) =⇒ f(x+∆x)− f(x)∆x

< 0

still!

Either way,


< 0 =⇒ f′(x) = lim∆x→0


≤ 0


. . . . . .




f(x+∆x) < f(x) =⇒ f(x+∆x)− f(x)∆x

< 0


f(x+∆x) > f(x) =⇒ f(x+∆x)− f(x)∆x

< 0

still! Either way,


< 0 =⇒ f′(x) = lim∆x→0


≤ 0


. . . . . .

Another important derivative fact

FactIf the graph of f has a horizontal tangent line at c, then f′(c) = 0.

Proof.The tangent line has slope f′(c). If the tangent line is horizontal, itsslope is zero.


. . . . . .

Another important derivative fact

FactIf the graph of f has a horizontal tangent line at c, then f′(c) = 0.

Proof.The tangent line has slope f′(c). If the tangent line is horizontal, itsslope is zero.


. . . . . .

Outline



Other notations



. . . . . .

Differentiability is super-continuity

TheoremIf f is differentiable at a, then f is continuous at a.

Proof.We have

limx→a

(f(x)− f(a)) = limx→a

f(x)− f(a)x− a

· (x− a)

= limx→a

f(x)− f(a)x− a

· limx→a

(x− a)

= f′(a) · 0 = 0

Note the proper use of the limit law: if the factors each have a limit ata, the limit of the product is the product of the limits.


. . . . . .



Proof.We have

limx→a


f(x)− f(a)x− a

· (x− a)

= limx→a

f(x)− f(a)x− a

· limx→a

(x− a)

= f′(a) · 0 = 0



. . . . . .



Proof.We have

limx→a


f(x)− f(a)x− a

· (x− a)

= limx→a

f(x)− f(a)x− a

· limx→a

(x− a)

= f′(a) · 0 = 0



. . . . . .

Differentiability FAILKinks

. .x

.f(x)

. .x

.f′(x)

.

.


. . . . . .


. .x

.f(x)

. .x

.f′(x)

.

.


. . . . . .


. .x

.f(x)

. .x

.f′(x)

.

.


. . . . . .

Differentiability FAILCusps

. .x

.f(x)

. .x

.f′(x)


. . . . . .


. .x

.f(x)

. .x

.f′(x)


. . . . . .


. .x

.f(x)

. .x

.f′(x)


. . . . . .

Differentiability FAILVertical Tangents

. .x

.f(x)

. .x

.f′(x)


. . . . . .


. .x

.f(x)

. .x

.f′(x)


. . . . . .


. .x

.f(x)

. .x

.f′(x)


. . . . . .

Differentiability FAILWeird, Wild, Stuff

. .x

.f(x)

This function is differentiable at0.

. .x

.f′(x)

But the derivative is notcontinuous at 0!


. . . . . .

Differentiability FAILWeird, Wild, Stuff

. .x

.f(x)

This function is differentiable at0.

. .x

.f′(x)

But the derivative is notcontinuous at 0!


. . . . . .

Outline



Other notations



. . . . . .

Notation

I Newtonian notation

f′(x) y′(x) y′

I Leibnizian notation

dydx

ddx

f(x)dfdx

These all mean the same thing.


. . . . . .

Link between the notations

f′(x) = lim∆x→0


= lim∆x→0

∆y∆x

=dydx

I Leibniz thought ofdydx

as a quotient of “infinitesimals”

I We think ofdydx

as representing a limit of (finite) differencequotients, not as an actual fraction itself.

I The notation suggests things which are true even though theydon’t follow from the notation per se


. . . . . .

Meet the Mathematician: Isaac Newton

I English, 1643–1727I Professor at Cambridge

(England)I Philosophiae Naturalis

Principia Mathematicapublished 1687


. . . . . .

Meet the Mathematician: Gottfried Leibniz

I German, 1646–1716I Eminent philosopher as

well as mathematicianI Contemporarily disgraced

by the calculus prioritydispute


. . . . . .

Outline



Other notations



. . . . . .


If f is a function, so is f′, and we can seek its derivative.

f′′ = (f′)′

It measures the rate of change of the rate of change!

Leibniziannotation:

d2ydx2

d2

dx2f(x)

d2fdx2


. . . . . .


If f is a function, so is f′, and we can seek its derivative.

f′′ = (f′)′

It measures the rate of change of the rate of change! Leibniziannotation:

d2ydx2

d2

dx2f(x)

d2fdx2


. . . . . .

function, derivative, second derivative

. .x

.y.f(x) = x2

.f′(x) = 2x

.f′′(x) = 2


. . . . . .

Summary

I A function can be differentiated at every point to find its derivativefunction.

I The derivative of a function notices the monotonicity of thefunction (fincreasing =⇒ f′ ≥ 0)

I The second derivative of a function measures the rate of thechange of the rate of change of that function.


Documents

Lesson 7: The Derivative as a Function