the derivative english

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LIST OF SLIDE

A. DEFINITION

B. THE DERIVATIVE FUNCTIONS

C. PARAMETRIC FUNCTION

2

D. IMPLICIT FUNCTION

E. APLICATION OF DERVATIVE



A. DEFINITION

3

What is the derivative ?



A. DEFINITION

4

In calculus (a branch of mathematics) thederivative is a measure of how a function

changes as its input changes. Loosely speaking, aderivative can be thought of as how much onequantity is changing in response to changes insome other quantity; for example, the derivative

of the position of a moving object with respect totime is the object's instantaneous velocity.

http://en.wikipedia.org/wiki/Calculus

http://en.wikipedia.org/wiki/Mathematics

http://en.wikipedia.org/wiki/Function_(mathematics)

http://en.wikipedia.org/wiki/Function_(mathematics)


http://en.wikipedia.org/wiki/Calculus



A. DEFINITION

5

The derivative of a function at a chosen input valuedescribes the best linear approximation of the functionnear that input value. For a real-valued function of asingle real variable, the derivative at a point equals theslope of the tangent line to the graph of the function atthat point. In higher dimensions, the derivative of afunction at a point is a linear transformation called thelinearization.[1] A closely related notion is the differential

of a function. The process of finding a derivative is calleddifferentiation..

http://en.wikipedia.org/wiki/Linear_approximation

http://en.wikipedia.org/wiki/Real-valued_function

http://en.wikipedia.org/wiki/Slope

http://en.wikipedia.org/wiki/Tangent_line

http://en.wikipedia.org/wiki/Graph_of_a_function

http://en.wikipedia.org/wiki/Linear_transformation

http://en.wikipedia.org/wiki/Linearization

http://d/downloads/Derivative.htm

http://en.wikipedia.org/wiki/Differential_of_a_function












http://en.wikipedia.org/wiki/Linearization














http://en.wikipedia.org/wiki/Slope











A. DEFINITION

6

It can be shown by the graphic bellow :

The graph of a function, drawn in black, and atangent line to that function, drawn in red. The slopeof the tangent line is equal to the derivative of thefunction at the marked point.





A. DEFINITION

7

Various Symbols for the Derivative

( ) or '( ) ordy df x f xdx dx

0

Definition: lim x

dy y

dx x



A. DEFINITION

8

Higher Order Derivatives

Let y = f ( x ). We have:



9

Some Basic Derivatives

A. DEFINITION



1

A. DEFINITION



11

A. DEFINITION

Chain Rule

The last formula

is known as the Chain Rule formula. It may be

rewritten as

Another similar formula is given by



B. THE DERIVATVE FUNCTIONS

12

Derivative of Trigonometry



13


1. = 4 + 2,

…

= 44 − 22

2. = 2 3 + 4,

…

= 2 23 + 4. 32

= 62. 23 + 4



14


erivative of the Hyperbolic functions and theirInverses



15




16


Recall the definitions of the trigonometricfunctions



17

Derivative of the Exponential and Logarithmic functions




18

EXAMPLE


= 24 + 3,

……?

Solution :

1)

Remember the Formula :()

= −1

(24 + 3)

= 2. ln4 + 3.(4 + 3

2) After that, differentiate( 4+3

, by using the formula :

(ln)

=1



19

B. THE DERIVATVE FUNCTIONS(ln(4+3))

=

1

(4+3). 3=

3

(4+3)

Finally, from the explanation above we can solve the ques tion

by :

24+3

= 2.ln4+3.(4+3

= 2. ln4+3.1

(4+3)4+ 3

= 2. ln4+3.3

(4+3)

=6 ln(4+3)(4+3)



2

Derivative of yclometry


In mathematics, the inverse trigonometric functions or cyclometric

functions are the inverse functions of the trigonometric functions.

The principal inverses are listed in the following table.


http://en.wikipedia.org/wiki/Inverse_function

http://en.wikipedia.org/wiki/Trigonometric_function

http://en.wikipedia.org/wiki/Trigonometric_function

http://en.wikipedia.org/wiki/Inverse_function




21

B. THE DERIVATVE FUNCTION

NameUsual

notationDefinition

Domain of x

for real result

Range ofusual

principal

value

(radians)

Range ofusual

principal

value

(degrees)

arcsine y = arcsin x x = sin y −1 ≤ x ≤ 1 −π/2 ≤ y ≤ π/2 −90° ≤ y ≤ 90°

arccosine y = arccos x x = cos y −1 ≤ x ≤ 1 0 ≤ y ≤ π 0° ≤ y ≤ 180°

arctangent y = arctan x x = tan y all real

numbers−π/2 < y < π/2 −90° < y < 90°

arccotangent y = arccot x x = cot y all real

numbers0 < y < π 0° < y < 180°

arcsecant y = arcsec x x = sec y x ≤ −1 or 1 ≤ x 0 ≤ y < π/2 or

π/2 < y ≤ π

0° ≤ y < 90° or

90° < y ≤ 180°

arccosecant y = arccsc x x = csc y x ≤ −1 or 1 ≤ x −π/2 ≤ y < 0

or 0 < y ≤ π/2

-90° ≤ y < 0°

or 0° < y ≤ 90°

http://en.wikipedia.org/wiki/Radian

http://en.wikipedia.org/wiki/Degree_(mathematics)

http://en.wikipedia.org/wiki/Sine

http://en.wikipedia.org/wiki/Cosine

http://en.wikipedia.org/wiki/Trigonometric_functions

http://en.wikipedia.org/wiki/Cotangent


http://en.wikipedia.org/wiki/Cosecant

http://en.wikipedia.org/wiki/Cosecant


http://en.wikipedia.org/wiki/Cotangent


http://en.wikipedia.org/wiki/Cosine

http://en.wikipedia.org/wiki/Sine

http://en.wikipedia.org/wiki/Degree_(mathematics)

http://en.wikipedia.org/wiki/Radian



23

EXAMPLE


= (2 − 5),……?

Answer :

=−1

1− (2 −5)2.(2 − 5)

=−1

1−(2

−10+25)

. 2

=−2

−2+10−24



24

C. PARAMETRIC DERIVATIVE

The common form of parametric function :

x = f(t)

y = g(t) (t as a parameter)

To differentiate the function in the form of parametric, take :

x = f(t), so

= ′

()

y = g(t), so

= ′()

It can be exlplained :()

()=

=

.

FORMULA x = f(t)

y = g(t)



25

if :

EXAMPLE

x = 2t2 + t

y = t5 + 1

find : a). b).




26

Solution

). = 22 + , = 4 + 1

= 5 + 1, = 54

=

= 5

4

4 + 1

). = 54,

22 = 203

= 4

+ 1,

2

2= 4

2 2

=

2 2

2 2

3 =

4+1203 .54.44+13 =

203(3+1)

4+13




27

Implicit erivativeIn many examples, especially the ones derived from differential

equations, the variables involved are not linked to each other in an

explicit way. Most of the time, they are linked through an implicitformula, like F ( x ,y ) =0. Once x is fixed, we may find y through

numerical computations. (By some fancy theorems, we may formally

show that y may indeed be seen as a function of x over a certain

interval). The question becomes what is the derivative , at least at a

certain a point? The method of implicit differentiation answers thisconcern. Let us illustrate this through the following example.

D. IMPLICIT DERIVATIVE



28

D. IMPLICIT DERIVATIVE

If there is implicit function : 3 + + 3 = 0, … ?

Solution :

3 + + 3 = 0

32 + + (.

+ 32 .

) = 0

+ 32.

= −(32 + )

=−32 +

+ 32



29

E. APLICATION DERIVATIVE

A patient is being fed inntravenously from a conical-shapedbottle of maximum radius 4 inchies and 10 inchies. If the

rate of flow is 28 cubic inchies per hour find the rate at

which the fluid level is dropping in the bottle when the level

is at 8 inchies; when the level is at 2 inchies.

Solution

We again follow the procedure outline in Remark 2.

1. The units indicated in the problem are inchies and hours.

From the sketch in Fig. 1 and the cross section in Fig 2 wechoose the variables r=radius of fluid in cone and h= height

of fluid in cone.



3

E. APLICATION DERIVATIVE

4

10 10

h

r

4

Fig. 1 Fig 2



31

E. APLICATION DERIVATIVE2. In term of these variables, we need to find

, the

rate at which the height of the fluid is changing. The

rate of flow is given to be :

= −28

Where V=volume of fluid in the cone.

3. The volume of fluid is given by the formula for the

volume of a cone.

=2

3

One of the variables can be eliminated by the

relationship between corresponding parts of similar

triangle in Fig.2



32

E. APLICATION DERIVATIVEThat is,

4=

10

Substituting = 0,4, we have

=

(0,4)2

3 =

0,163

3

So that

=

0,16(32)

3= 0,162

4. From the chain rule :

=

E APLICATION DERIVATIVE



33

E. APLICATION DERIVATIVEAnd so,

=

=

−28

0.162

In particular, when = 8,

=8 =

−28

0,16(8)2 =−0,87( )

When h = 2

=2 =−28

0,16(2)2= −13,93( )

Notice that the fluid level is dropping much more rapidly as the cone empties

even though the flow rate is a constant 28 cubic inchies per hour. You can

observe this phenomenon by watching as it empties.



34

BIBLIOGRAPHY

http://en.wikipedia.org/wiki/Derivative

http://www.sosmath.com/.../derivative.html

http://www.sosmath.com/.../der05.html

http://en.wikipedia.org/wiki/Inverse_trigonometric

L. Mett, Corren & C. Smith, James. 1991. Calculus With

Aplication . York Graphich Services, Inc. USA.









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