101- Full Derivative Report

7/26/2019 101- Full Derivative Report

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DerivativesCalculus I

Amro Ismail Kasht ..200802124 .. 19 Mohamed

Al-Khateeb .. 200702985 ..12 Mohamed Al-Qahtai ..

200!01884 .. 8 Mahm"d#aio"me .. 200801520 .. 1$


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The derivative

Theorem:

Let xdomf

1)

t x+f( t)f(x )

txlim

, if exists , is called the right (hand) derivative

of f at x

2)

t xf( t)f(x )

txlim

, if exists , is called the left (hand) derivative

of f at x

3) limt x

f( t) f(x )tx ,if exists , is called the derivative of

f at x

The derivative of f at x exists only if oth the right and left

derivatives are exist and e!ual"

#xam$le:

Letf(x )={3x , x 26 , x


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t 266t2

=0

t 2f( t)f(2)

t2 =lim

lim

&o the left derivative of f at 2 is e!ual to '

&ince the right derivative of f at 2 the left derivative of f at

2, then the derivative of f at 2 does not exist

2" %t x>2

limt x

f( t)f(x)tx

=limt x

3 t3xtx

=3

&o f'(x )=3 , x>2

3" %t x


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Then f (x) + n xn-1

%nd if f(x) + x- n

Then f (x) + (- n) x

- n . 1

#xam$le: /ind the 0rst derivative for the folloing

functions"

1)f(x) + x1

f (x) + 1 x'+ 1

2)f(x) + x2

f (x) + 2 x1+ 2 x

3)f(x) + x'

f(x) + 1

f (x) + '

)f(x) + x123456

f (x) + 123456

x123455

To con0rm using the

de0nition

limt x(

t

2

x

2

tx)=limt x(

(t

x ) (t+

x )tx )

+ x 7 x + 2x

8ememer: The

derivative of any


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)f(x) + x- 12

f (x) + -12 x-13

)f(x) + x

f(x) + x 12

f (x) +1

2 x . ;+

1

2x

4)f(x) +1

x

f(x) + x- 1

f (x) + -1 x-2+1

x2

5)f(x) +1

3

x2

f(x) + x2/3

f (x) + 23

x5 /3 + 2

33

x5

8ememer:

nx + x

1

n

nx

m

+ xm

n

1

nx + x

1

n

1

nxm + x

mn

1

xn + x

n


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%lgera of g) (x) + f (x) > g (x)

(f " g) (x) + f (x) " g (x) 7 g (x) " f (x)

(fg) (x) +g (x ) . f (x )g (x ) . f(x)

g2(x )

#xam$le: 0nd the 0rst derivative for the folloing

functions:

1)2 f (x) + x27 x 7 3

2 f (x) + 2x 7 1

f (x) +2x+1

2

2)f (x) + (x7 2x 7 6) (2x4. x7 x27 3)


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f (x) + (x7 2x 7 6) (1x. x7 5x) 7 (2x4. x7 x27 3)(x27 2)

3)f (x) +x

2

x72x+3

f (x) +(x72x+3)(2x )(7x62 )(x2)

(x72x+3 )2

)f(x) +x

8+53x

9+x+3

f (x) +(3x9+x+3) (8x7 )(27x8+1)(x8+5)

(3x9+x+3 )2


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The derivative of the com$osite function

Theorem (1):

Let y= f(u ) ,u=g (x ) , g continuous at x0 and f continuous at

u0=g (x)

Let f'(u0) e the derivative of f res$ect to u at u0 and g

'(x0)

the derivative of g ith res$ect to " If f' (uo ) and g

'(xo )R ,then:

(fog )'(xo )=f

'(uo ). g'(xo)=f

'(g (xo )) . g'(xo)

This formula is ?non as @the chair ruleA

#xam$le:

Let f(x )=(7x3+2x22x+5)

9

Then f' (x )=9 (7x3+2x22x+5 )8 .(21x2+4x2 )

Theorem (2):

Let y= f'(x )=xn , nQ{0 }

Then y'=f'(x )=n xn1

#xam$le:


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f(x )=

1

3(2x3+x3 )5=(2x3+x3 )

512

f'

(x )=512 .(2x

3

+x3 )

1712

. (6x2

+1)= 5 (6x2+1 )

12 .12

(2x3+x3)17

Im$licit di=erentiation

% function or (more than one function) maye given im$licitly y the

mean of an e!uation" In this case, to 0nd the derivative of the function,

e ta?e advantage of theorem (1), as shon in the folloing exam$le

#xam$le:

%ssuming that y re$resent a function (or more) in , 0nd the

derivative y ' of y ith res$ect to x

x2+y2=25 y

2=25x2 y=25x2

2x+2y . y'=0

2y . y'=2x

y

'=2x2y

=xy


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If e gra$h x2+y2=25 : e get a circle ith radius +

The Derivative as the Instantaneous Rate

Of change


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Let y be a variable that depends on another variable x. let y be elated to x according to the

function: y = f(x

If x change fro! xo to t" y changes fro! f (xo to f (t. Let#s denote the difference

t$xo by

x and the difference f(t % f(&o by

y . thus "'hen x change by !ount

x

y change by the a!ount

y.

the uotientx

y

is called the average nate of change y 'ith respect to x on the interval

)xo " t *.

Thexot li!

x

y

is called the instantaneous nate of change at xo of y 'ith respect to

x.

but li! ay+ ax =xot

Lim

Xot

Xoftf

((

=

#f

(&o


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thus the instantaneous nate of change of y = f(x at xo 'ith respect to x is eual to

#f

&o .(

This result has !any applications in various scientific disciplines

If , (t " v (t and a(t are respectively " the position" the velocity and the acceleration of a

!oving ob-ect as function of ti!e t then at any instant t

" 'e have

(t

=#s(t

and a (t

=#v(t

/xa!ple:

Let s (t =

0

0

1t

$

2

2

3t

4 5t 6 t

7

8ind the follo'ing:

1$a for!ula of v(t and a(t as function of t

2$the instance t 'hich the ob-ect is te!porally at halt

0$the instance at 'hich the ob-et experience no acceleration . 9hats its

velocity at !o!ent.

5$'hat is the velocity of ob-et at t= 2 and t= 3

3$'hat is acceleration of ob-et at t= 2 and t= 0

,olution:

71$(t= t

2

$ 3t 4 5 = (t$1(t$5 6 t


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7a(t = 2t % 3 = 2(t $2

3

6 t

2$v(t = 7 ; t


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Higher Derivatives

Higher derivatives are obtained by repeatedly differentiating anexpression or equation.

First derivative:

Second derivative:

This is obtained by differentiating y with respect to x twice.Third derivative:

This is obtained by differentiating y with respect to x thrice.

Obtaining higher derivatives is relative simple. Consider the equation


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y x!" #x$" %x#&$x & '

Example :

If " then 'hat are the higher derivatives?

@ns'er

The first derivative:

Ao' for the second derivative. 9e -ust differentiate our previous ans'er.

Ao' for the 0rd and 5th derivatives.
http://www.intmath.com/Differentiation/A9-A.htmhttp://www.intmath.com/Differentiation/A9-A.htm


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Ao' the 3th derivative is

yv= 127.

The Bth" Cth" th and all other derivatives are 7" since the derivative of a constant is 7.

/xa!ple 2:

Let f(x = x

3

. find f

(n

(x 6 x

IA.

,olution:

f

1(

(x = 3 x

5

f

2(

(x = 3. 5. x

0

f

0(

(x = 3.5.0. x

2

f

5(

(x = 3.5.0.2. x

f

3(

(x = 3.5.0.2.1

f

(k

(x = 7 6 E

IA

B

f

(n

(x =

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101- Full Derivative Report