Ex_2_6_FSC_part2
13
mathcity.org Merging man and maths !"#$%&" ()* +,-./0%-1&2 Calculus and Analytic Geometry, MATHEMATICS 12 Available online @ http://www.mathcity.org, Version: 1.0.0 2.10 Derivative of General Exponential Function (Page 80) A fun cti on def in e by ( ) x x a = where 0 , 1 a a > ¹ is called general exponential function. Suppose x y a = Þ x x y y a d d + + = Þ x x a y d d + = - Þ x x x a a d d + = - Since x y a = Þ ( 1 ) x x y a a d d = - Dividing by d ( 1 ) x x y a a x x d d d d - = Taking limit as 0 d ® 0 0 ( 1 ) lim lim x x x x y a a x d d d d d d ® ® - = Þ 0 1 lim x x x dy a a dx x d d d ® æ ö ç ÷ ç ÷ è ø - = Þ 0 1 lim x x x dy a a dx x d d d ® æ ö ç ÷ ç ÷ è ø - = Þ ( ) . ln x x d a a a dx = Since 0 1 l im l n x x a a x ® - = Derivative of Natural Exponential Function Th e expon ential function ( ) x f x e = where 2.71828... e = is called Natural Exponential Function. Suppose x e = Do yourself … Just Change a by e in above article. You’ll get x x d e e dx = 2.11 Derivative of General Logarithmic Function (page 81) If 0, 1 a a > ¹ and a = , then the functi on defined by log ( 0) a x x = > is called General Logarithmic Function. Suppose log a x = Þ log ( ) a y x x d d + = + Þ log ( ) a x x y d d = + - Þ log( ) log a a y x x x d d = + - log a x x x d æ ö ç ÷ è ø + = Since log log log a a a m m n n - = Dividing both sides by x d 1 log a y x x x x x d d d d æ ö ç ÷ è ø + = Taking limit as 0 d ® 0 0 1 l im lim log a x x y x x x x x d d d d d d ® ® æ ö ç ÷ è ø + = Þ 0 1 lim log 1 a x dy x dx x x d d d ® æ ö ç ÷ è ø = + 1 x x d æ ö
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Transcript of Ex_2_6_FSC_part2
!"#$%&" ()* +,-./0%-1&2 Calculus and Analytic
Geometry, MATHEMATICS 12
Available online @ http://www.mathcity.org, Version: 1.0.0
2.10 Derivative of General Exponential Function (Page 80) A function define by ( ) x x a= where 0, 1a a> ¹
is called general exponential function.
Suppose x y a=
++ = Þ x xa yd d
+= -
+= - Since x y a=
Dividing by d
x x
d d
d d
- = Þ
0
- =
dx = Since
- =
Derivative of Natural Exponential Function The exponential function ( ) x f x e= where 2.71828...e = is called Natural Exponential
Function.
Suppose xe=
Do yourself … Just Change a by e in above article. You’ll get
x xd e e
dx =
2.11 Derivative of General Logarithmic Function (page 81) If 0, 1a a> ¹ and a= , then the function defined by log ( 0)a x x= > is
called General Logarithmic Function.
Suppose loga x=
Þ log ( )a y x xd d + = + Þ log ( )a x x yd d = + -
Þ log ( ) loga a y x x xd d = + -
loga
m m n
1 loga
+ =
y x x
+ =
dy x
= +
dy x x
= +
= + Since log log m a am x x=
Þ 0
®
= +
e ®
dx x a = Since log lne a a=
Derivative of Natural Logarithmic Function The logarithmic function ( ) loge x x= where 2.71828...e = is called Natural
Logarithmic Function. And we write ln x instead of loge x for our ease.
Suppose ln x=
Þ ln( ) y x xd d + = + Þ ln ( ) x x yd d = + -
Þ ln( ) ln x x xd d = + -
Þ ln x x
m m n
1 1
y x
= +
dy x x
Þ 0
dy x x
= +
= + Since ln ln mm x x=
Þ 0
®
= +
e ®
Þ ( ) 1
ln d
Diff. w.r.t
dx dx
dx
-¢Þ = -
Diff. w.r.t 1
dx dx =
¢Þ = +
d e e x
x
dx x dx x
1
Question # 1(iii)
Diff. w.r.t
dx dx = +
( ) ( )( ) 1 ln 1 ln x xd d f x e x x e
dx dx ¢Þ = + + +
æ ö = + + +ç ÷
x
or ( )1 1 ln
x
dx dx e-
æ ö = ç ÷
dx dx f x e
- -
-
dx dx
x x
x x
dx dx e e
ax ax
d d e e e e e e e e
dx dx
e e
ax ax
e e e a e a e e e e a
e e
ax ax
a e e e e a e e e e
e e
e e
+
( )
( 2 ) ( 2 ) ax ax ax ax ax ax ax ax
ax ax
e e
ax ax
e e
( ) 2
4 ( )
( )2 2 1 2
dx dx
( ) ln ln 2
x x x xd f x e e e e
dx
( ) ( ) ( )2 2
e ee e
( )2 2 1 2ln
x x e e
2
x x f x e e-Þ = + ln lnm x m x=Q
Now diff. w.r.t x
dx dx
ln 2 x xÞ = ln lnm x m x=Q
Now diff. w.r.t x
dx dx
d d x x x x
dx dx = +
2
dx
dx dx
2 1 ln y x
x = 2 1ln y x x-Þ = 2 ln y x xÞ = -
Now do yourself.
2
+è ø
22 2
2 1 1
x dx dx
( ) ( )
( )
( )
2
Diff. w.r.t
æ ö ç ÷= + ×ç ÷
1
2 sin2 xdy d e x
dx dx
-Þ =
2 2sin 2 sin 2 x xd d e x x e
dx dx
- -= +
2 2cos2 (2) sin2 ( 2) x xe x x e- -= + - ( )2 2 cos2 sin 2
x e x x
-= + +
dx dx
-= + +
( ) ( )3 2 3 22 1 2 1 x xd d e x x x x e
dx dx
x x e x x x x e
- -= + + + + + × -
x x e x x x x e
- -= + - + + × ( )2 3 2 3 4 2 1
x e x x x x
-= + - - -
-= - + + - Answer
Diff w.r.t x
dx dx = +
( )sin cos 1 xe x x= + Answer
Question # 2(xi) Do yourself
= +
x x x dx dx dx
Þ = + + +
( ) ( ) 1
d x x x
( ) ln
Diff w.r.t x
y x x dx dx
= ×
x x x y dx dx dx
Þ = +
d x x x
dx x
x xdy x
( )2
( )21 3 ln ln( 1) ln 1
2 2 y x x xÞ = - - - +
Now diff. w.r.t x
2 1 2 1
y dx x dx dx x x Þ = - - - +
- - +
( ) ( ) ( )
x x x = - -
2 1 1
dx x x x
é ù- + - - - ê úÞ =
2 1 11
x x x x x
é ù- + - - - +- ê ú= ×
- - +ê ú- + ë û
2 1 1
x x x +-
( ) ( )
Do yourself
2.1.3 Derivative of Hyperbolic Function (page 85) The hyperbolic functions are define by
sinh , 2
x e e
{ } 1 2
= = Î - -
= = Î +
e e
sinh 2 2 2
x x x x x xd d e e d d
e e e e dx dx dx dx
- - -æ ö- æ ö
1 2 2 2
x x x x x xd d e e e e e e
dx dx
è ø
(iv) coth x x
dx dx e e
x x
d d e e e e e e e e
dx dx
e e
( 1) ( 1) x x x x x x x x
x x
e e
x x
e e
( 2 ) ( 2 ) x x x x x x x x
x x
e e
x x
( )
e ee e --
- æ ö = = - = -ç ÷
x x
dx dx e e dx dx
- -- -
-
2 1 x x x xd e e e e
dx
( ) ( )( ) ( )
x x e e e e e e
e e
x x x x x x x x
e e e e
- -
- - - -
- - - = = -
+ + + +
differentiate w.r.t. x.
sinh d d
x dx dx
Þ 2
(ii) Do yourself as above.
(iii) Do yourself as (iv) below or see book at page 88.
(iv) Let 1coth x-= Þ coth y x=
differentiate w.r.t. x
coth d d
Þ
differentiate w.r.t. x
sech d d
y y dx
Þ 2
Question # 3(i) cosh 2 y x=
Diff. w.r.t
x dx dx
Þ = 2sinh2 dy
2
2
3
Made by: Atiq ur Rehman ([email protected] ) , http://www.mathcity.org
2 2cosh sinh 1q q - =Q 2 21 tanh sechq q \ - =
Available online @ http://www.mathcity.org, Version: 1.0.0
2.10 Derivative of General Exponential Function (Page 80) A function define by ( ) x x a= where 0, 1a a> ¹
is called general exponential function.
Suppose x y a=
++ = Þ x xa yd d
+= -
+= - Since x y a=
Dividing by d
x x
d d
d d
- = Þ
0
- =
dx = Since
- =
Derivative of Natural Exponential Function The exponential function ( ) x f x e= where 2.71828...e = is called Natural Exponential
Function.
Suppose xe=
Do yourself … Just Change a by e in above article. You’ll get
x xd e e
dx =
2.11 Derivative of General Logarithmic Function (page 81) If 0, 1a a> ¹ and a= , then the function defined by log ( 0)a x x= > is
called General Logarithmic Function.
Suppose loga x=
Þ log ( )a y x xd d + = + Þ log ( )a x x yd d = + -
Þ log ( ) loga a y x x xd d = + -
loga
m m n
1 loga
+ =
y x x
+ =
dy x
= +
dy x x
= +
= + Since log log m a am x x=
Þ 0
®
= +
e ®
dx x a = Since log lne a a=
Derivative of Natural Logarithmic Function The logarithmic function ( ) loge x x= where 2.71828...e = is called Natural
Logarithmic Function. And we write ln x instead of loge x for our ease.
Suppose ln x=
Þ ln( ) y x xd d + = + Þ ln ( ) x x yd d = + -
Þ ln( ) ln x x xd d = + -
Þ ln x x
m m n
1 1
y x
= +
dy x x
Þ 0
dy x x
= +
= + Since ln ln mm x x=
Þ 0
®
= +
e ®
Þ ( ) 1
ln d
Diff. w.r.t
dx dx
dx
-¢Þ = -
Diff. w.r.t 1
dx dx =
¢Þ = +
d e e x
x
dx x dx x
1
Question # 1(iii)
Diff. w.r.t
dx dx = +
( ) ( )( ) 1 ln 1 ln x xd d f x e x x e
dx dx ¢Þ = + + +
æ ö = + + +ç ÷
x
or ( )1 1 ln
x
dx dx e-
æ ö = ç ÷
dx dx f x e
- -
-
dx dx
x x
x x
dx dx e e
ax ax
d d e e e e e e e e
dx dx
e e
ax ax
e e e a e a e e e e a
e e
ax ax
a e e e e a e e e e
e e
e e
+
( )
( 2 ) ( 2 ) ax ax ax ax ax ax ax ax
ax ax
e e
ax ax
e e
( ) 2
4 ( )
( )2 2 1 2
dx dx
( ) ln ln 2
x x x xd f x e e e e
dx
( ) ( ) ( )2 2
e ee e
( )2 2 1 2ln
x x e e
2
x x f x e e-Þ = + ln lnm x m x=Q
Now diff. w.r.t x
dx dx
ln 2 x xÞ = ln lnm x m x=Q
Now diff. w.r.t x
dx dx
d d x x x x
dx dx = +
2
dx
dx dx
2 1 ln y x
x = 2 1ln y x x-Þ = 2 ln y x xÞ = -
Now do yourself.
2
+è ø
22 2
2 1 1
x dx dx
( ) ( )
( )
( )
2
Diff. w.r.t
æ ö ç ÷= + ×ç ÷
1
2 sin2 xdy d e x
dx dx
-Þ =
2 2sin 2 sin 2 x xd d e x x e
dx dx
- -= +
2 2cos2 (2) sin2 ( 2) x xe x x e- -= + - ( )2 2 cos2 sin 2
x e x x
-= + +
dx dx
-= + +
( ) ( )3 2 3 22 1 2 1 x xd d e x x x x e
dx dx
x x e x x x x e
- -= + + + + + × -
x x e x x x x e
- -= + - + + × ( )2 3 2 3 4 2 1
x e x x x x
-= + - - -
-= - + + - Answer
Diff w.r.t x
dx dx = +
( )sin cos 1 xe x x= + Answer
Question # 2(xi) Do yourself
= +
x x x dx dx dx
Þ = + + +
( ) ( ) 1
d x x x
( ) ln
Diff w.r.t x
y x x dx dx
= ×
x x x y dx dx dx
Þ = +
d x x x
dx x
x xdy x
( )2
( )21 3 ln ln( 1) ln 1
2 2 y x x xÞ = - - - +
Now diff. w.r.t x
2 1 2 1
y dx x dx dx x x Þ = - - - +
- - +
( ) ( ) ( )
x x x = - -
2 1 1
dx x x x
é ù- + - - - ê úÞ =
2 1 11
x x x x x
é ù- + - - - +- ê ú= ×
- - +ê ú- + ë û
2 1 1
x x x +-
( ) ( )
Do yourself
2.1.3 Derivative of Hyperbolic Function (page 85) The hyperbolic functions are define by
sinh , 2
x e e
{ } 1 2
= = Î - -
= = Î +
e e
sinh 2 2 2
x x x x x xd d e e d d
e e e e dx dx dx dx
- - -æ ö- æ ö
1 2 2 2
x x x x x xd d e e e e e e
dx dx
è ø
(iv) coth x x
dx dx e e
x x
d d e e e e e e e e
dx dx
e e
( 1) ( 1) x x x x x x x x
x x
e e
x x
e e
( 2 ) ( 2 ) x x x x x x x x
x x
e e
x x
( )
e ee e --
- æ ö = = - = -ç ÷
x x
dx dx e e dx dx
- -- -
-
2 1 x x x xd e e e e
dx
( ) ( )( ) ( )
x x e e e e e e
e e
x x x x x x x x
e e e e
- -
- - - -
- - - = = -
+ + + +
differentiate w.r.t. x.
sinh d d
x dx dx
Þ 2
(ii) Do yourself as above.
(iii) Do yourself as (iv) below or see book at page 88.
(iv) Let 1coth x-= Þ coth y x=
differentiate w.r.t. x
coth d d
Þ
differentiate w.r.t. x
sech d d
y y dx
Þ 2
Question # 3(i) cosh 2 y x=
Diff. w.r.t
x dx dx
Þ = 2sinh2 dy
2
2
3
Made by: Atiq ur Rehman ([email protected] ) , http://www.mathcity.org
2 2cosh sinh 1q q - =Q 2 21 tanh sechq q \ - =