!"#$%&" ()* +,-./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 (mathcity@gmail.com ) , http://www.mathcity.org
2 2cosh sinh 1q q - =Q   2 21 tanh sechq q \ - =