In this chapter, we begin our study of differential calculus.

In this chapter, we begin our study of differential calculus.

This is concerned with how one quantity changes in relation to another quantity.

the derivative is a measure of how fast does a function change in response to changes in independent variable; for example, the derivative of the position of a moving object with respect to time is the object's velocity.

DERIVATIVES

Definition of INFINITESIMAL

1: taking on values arbitrarily close to but greater than zero 2: immeasurably or incalculably small <an infinitesimal difference>

Now concentrate and observe:

x

P

T

Let’s assume that we know function at any → graph.

Let’s find the slope of the line joining position P(x = a) and some T .

𝑓 (𝑎+∆𝑥 ) − 𝑓 (𝑎)∆𝑥 =

𝑎 𝑎+∆𝑥

𝑓 (𝑥)f (x+Δx) - f(x)

Let us consider Δ x intervals that are getting smaller and smaller.

x

f (x

)

D x

Df = f (x+Δx) - f(x)

P

QR

S T

D x

f (x+Δx) - f(x)

ALTHOUGH, both, Δ f and Δ x are becoming infinitesimally small approaching zero, their ratio is approaching definite value – slope of tangent line at PSlopes of straight lines connecting point P and other points on the path are approaching the slope of the line tangent at P.

lim∆ 𝑥→0

𝑓 (𝑎+∆ 𝑥 )− 𝑓 (𝑎)∆ 𝑥

𝑎

Explain yourself and me what you saw

The derivative of a function at a fixed number is

lim∆ 𝑥→0

𝑓 (𝑎+∆ 𝑥 )− 𝑓 (𝑎)∆ 𝑥

provided that this limit exists.

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

Very often you are going to find that definition in following form:

Graphical interpretation of mathematical definition of derivative at point is the slope of the tangent line to the function at point .

𝑓 ′ (𝑎)=¿

limh →0

𝑓 (𝑎+h )− 𝑓 (𝑎)h

𝑓 ′ (𝑎)=¿

Till now, we considered the derivative of a function at a fixed number .Now, we change our point of view and let the number vary. Let’s assume that can take any value of on an open interval

The derivative is the instantaneous rate of change of a function with respect its variable. This is equivalent to finding the slope of the tangent line to the function at a point.

provided this limit exists.If this limit exists for each x in an open interval I, then we say that f is differentiable on I.

Definition: Let be a function. The derivative of is the function whose value at x is the limit

limh →0

𝑓 (𝑥+h ) − 𝑓 (𝑥)h

𝑓 ′ (𝑥 )=¿

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

y “y prime”

dydx

“d y d x” or “the derivative of y with respect to x”

dfdx

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

d f xdx “d dx of f of x” or “the derivative of f of x with respect to x”

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

0limx

dy ydx xD

D

D

• This means that: – When the derivative is large (and therefore the curve

is steep, as at the point P in the figure), the y-values change rapidly. – When the derivative is small, the curve is relatively flat and the y-

values change slowly.

Rates of change

y f x

y f x

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

The derivative is the slope of the original function.

𝑥

𝑥

2 3y x

2 2

0

3 3limh

x h xy

h

2 2 2

0

2limh

x xh h xyh

2y x

0𝑦 ′=limh → 0

(2𝑥+h )

A function is differentiable if it has a derivative everywhere in its domain.

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

Functions on closed intervals must have one-sided derivatives defined at the end points.

Differentiability

Derivatives will fail to exist at:

1, 0 1, 0

xf x

x

f x x

3f x x

23f x x

corner cusp

vertical tangent discontinuity

A function has derivative at a point, if the left derivative is equal to the right derivative at that point.

The left slope must be equal to the right slope.

Find an equation of the tangent line to the hyperbola at the point (3, 1).

The slope of the tangent at (3, 1) is:

Eq. of the tangent at the point (3, 1) is

x + 3y – 6 = 0

The hyperbola and its tangent are shown in the figure

How do you find normal at (3,1) ?

Equation of the tangent line to a function at the point

Equation of the normal line to a function at the point

𝑦− 𝑓 (𝑎 )= 𝑓 ′ (𝑎 )(𝑥−𝑎)

𝑦− 𝑓 (𝑎 )=− 1𝑓 ′ (𝑎 )

(𝑥−𝑎)

We can estimate the value of the derivative at any value of x by drawing the tangent at the point (x,f(x)) and estimating its slope.

For instance, for x = 5, we draw the tangent at P in the figure and estimate its slope to be about 3/2, so . This allows us to plot the point P’(5, 1.5) on the graph of f’ directly beneath P.

Repeating this procedure at several points, we get the graph shown in this figure.

Tangents at x = A, B, and C are horizontal.

So, the derivative is 0 there and the graph of f’ crosses the x-axis at those points.

Between A and B, the tangents have positive slope. So, f’(x) is positive there.

Between B and C, and the tangents have negative slope. So, f’(x) is negative there.

3 3

0 0

( ) ( )'( ) lim limh h

x h x h x xf x h f xf xh h

3 2 2 3 3

0

3 3limh

x x h xh h x h x xh

23 1x

example: Find Sketch both graphs.

Notice that = 0 when f has horizontal tangents and is positive when the

tangents have positive slope.

• We use a graphing device to graph f • and f’ in the figure.

– Notice that f’(x) = 0 when f has horizontal tangents and f’(x) is positive when the tangents have positive slope.

– So, these graphs serve as a check on our work in part (a).

THE DERIVATIVE AS A FUNCTION Example 2 b

Two theorems:

If f is differentiable at x = c, then f is continuous at x = c.

Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

)('lim)('lim xfxfcxcx

)()(lim)(lim cfxfxfcxcx

1. Differentiability Implies Continuity

The converse: "If a function is continuous at c, then it is differentiable at c," - is not true.

This happens in cases where the function "curves sharply."

𝑣𝑖𝑠𝑢𝑎𝑙𝑙𝑦 : 𝑙𝑒𝑓𝑡 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 h𝑡 𝑒𝑠𝑙𝑜𝑝𝑒≠ h𝑟𝑖𝑔 𝑡 𝑙𝑖𝑚𝑖𝑡𝑜𝑓 h𝑡 𝑒𝑠𝑙𝑜𝑝𝑒

Differentiability implies continuity, continuity doesn’t imply differentiability.

Intermediate Value Theorem for Continuous Functions

If is continuous on and is any number between and , then there is at least one number such that .

example:

Prove that function has a root/zero between 2 and 2.5.

is continuous on ], and , so must have a zero between 2 and 2.5.

Between and , must take on every

value between ½ and 3.

Intermediate Value Theorem for Derivatives

If and are any two points in an interval on which is differentiable, then takes on every value between and .

You can find it in this form too:

Let be differentiable on and suppose that k is a number between and . Then there exists a point such that .

is continuous function on

How to find derivatives???

If it were always necessary to compute derivatives directly from the definition,

calculus BC would be even worse nightmare, there would be no Ipad and the life as we know it ……. Computations would be tedious, and the evaluation of some limits would require ingenuity.

Fortunately, several rules have been developed for finding derivatives without having to use the definition directly.

These formulas greatly simplify the task of differentiation.

Let’s start with the simplest of all functions — the constant function f(x) = c.

First let us see what are we avoiding:

𝑓 (𝑥 )=𝑐𝑓 ′ (𝑥 )=lim

h → 0

𝑓 (𝑥+h )− 𝑓 (𝑥 )h

=limh→ 0

𝑐−𝑐h

=limh → 0

0

𝑓 ′ (𝑥 )=0The graph of this function is the horizontal

line y = c, which has slope 0.

𝑑𝑑𝑥 (𝑐 )=0

In Leibniz notation

The derivative of a constant is zero

examples: 3y

0y

2 22

0limh

x h xd xdx h

2 2 2

0

2limh

x xh h x

h

2x

3 33

0limh

x h xd xdx h

3 2 2 3 3

0

3 3limh

x x h xh h x

h

23x

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

(Pascal’s Triangle)

2

4d xdx

4 3 2 2 3 4 4

0

4 6 4limh

x x h x h xh h x

h

34x

2 3

We observe a pattern: 2x 23x 34x 45x 56x …

𝑓 (𝑥 )=𝑥𝑛 ,𝑛𝑖𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑖𝑛𝑡𝑒𝑔𝑒𝑟

examples:

4f x x

34f x x

8y x

78y x

We observe a pattern: 2x 23x 34x 45x 56x …

𝑑𝑑𝑥 (𝑥𝑛)=𝑛𝑥𝑛−1

power rule

examples:

5 4 47 7 5 35d x x xdx

limit has no effect on a constant coefficient; the constant could be factored to the outside.

𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑟𝑢𝑙𝑒 :𝑔 (𝑥 )=𝑐𝑓 (𝑥)

𝑓 ′ (𝑥 )=𝑐𝑔 ′ (𝑥)

𝑑𝑑𝑥 [𝑐 𝑓 (𝑥) ]=𝑐 𝑑

𝑑𝑥 𝑓 (𝑥)

4 12y x x

34 12y x

4 22 2y x x

34 4dy x xdx

𝑠𝑢𝑚𝑎𝑛𝑑𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑟𝑢𝑙𝑒

( ) ( ) ( ) ( )d d df x g x f x g xdx dx dx

limit of the sum = sum of the limitsprovided they are both differentiableeach term is treated separately

examples:

[ 𝑓 (𝑥 )+𝑔 (𝑥 ) ]′= limh → 0

𝑓 (𝑥+h )+𝑔 (𝑥+h ) − 𝑓 (𝑥 ) −𝑔 (𝑥)h

❑

¿ limh→0

¿¿

¿ limh →0

𝑓 (𝑥+h ) − 𝑓 (𝑥 )h

+ limh →0

𝑔 (𝑥+h ) −𝑔 (𝑥)h

example:

Example:Find the horizontal tangents of: 4 22 2y x x

Horizontal tangents occur when slope = zero.

2, 1, 1y y y

(The function is even, so we only get two horizontal tangents.)

𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑟𝑢𝑙𝑒

( ) ( ) ( ) ( ) ( ) ( ) d d df x g x f x g x g x f xdx dx dx

The derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function,provided they are both differentiable.

Example:

Find derivative if

Later, though, we will meet functions, such as y = x2 sinx, for which the product rule is the only possible method.

Q

Q

2

( ) ( ) ( ) ( )( )( ) ( )

d dg x f x f x g xd f x dx dxdx g x g x

The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator,provided they are both differentiable.

The theorems of this section show that:

Any polynomial is differentiable on .Any rational function is differentiable on its domain.

Furthermore, the Quotient Rule and the other differentiation formulas enable us to compute the derivative of any rational function—as the next example illustrates.

Example:

For instance:

It is possible to differentiate the function

using the Quotient Rule.However, it is much easier to perform the division first and write the function as

before differentiating.

Don’t use the Quotient Rule every time you see a quotient.

Sometimes, it’s easier to rewrite a quotient first to put it in a form that is simpler for the purpose of differentiation.

The Quotient Rule can be used to extend the Power Rule to the case where the exponent is a negative integer.

If n is a positive integer, then

1( ) n nd x nxdx

𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑃𝑜𝑤𝑒𝑟 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠

Example:

𝑃𝑜𝑤𝑒𝑟 𝑅𝑢𝑙𝑒

So far, we know that the Power Rule holds if the exponent n is a positive or negative integer.

If n = 0, then x0 = 1, which we know has a derivative of 0.

Thus, the Power Rule holds for any integer n.

What if the exponent is a fraction?

In fact, it can be shown by using Chain Rule (obviously proof later) that it also holds for any real number n.

If n is any real number, then

1( )n nd x nxdx

3 2

2/3

(2 /3) 123

5/323

1

( )

Let

Then

yx

dy d xdx dx

x

x

Example:

Example:

Find equations of the tangent line and normal line to the curve

at the point (1, ½). y= √𝑥1+𝑥2

In your mind:

slope of the tangent line at (1, ½) :

tangent line at (1, ½):

normal line at (1, ½):

𝑦−1=− 14

(𝑥− 1 )→ 𝑦=− 14 𝑥+

34

𝑦− 12=4 (𝑥− 1 )→ 𝑦=4 𝑥− 7

2

At what points on the hyperbola xy = 12 is the tangent line parallel to the line 3x + y = 0?

Since xy = 12 can be written as y = 12/x, we have:

Example:

𝑑𝑦𝑑𝑥=12 (𝑥−1 ) ′=− 12

𝑥2

Let the x-coordinate of one of the points in question be . Slope of the tangent line at that point is , and that has to be equal to the slope of line 3x + y = 0

the required points are: (2, 6) and (-2, -6)

Here’s a summary of the differentiation formulas we have learned so far.

1

'

2

0

' ' ' ' ' ' ' '

' '' ' '

n nd dc x nxdx dx

cf cf f g f g f g f g

f gf fgfg fg gfg g

𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛 𝐹𝑜𝑟𝑚𝑢𝑙𝑎𝑠

Higher Order Derivatives:

dyydx

is the first derivative of y with respect to x.

2

2

dy d dy d yydx dx dx dx

is the second derivative.

(y double prime)

dyydx

is the third derivative.

4 dy ydx

is the fourth derivative.

We will learn later what these higher order derivatives are used for.

p