JEE Exam  >  JEE Notes  >  Mathematics (Maths) for JEE Main & Advanced  >  NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives

NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

Download, print and study this document offline
Please wait while the PDF view is loading
 Page 1


Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 21: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers):   
? ? sin
cos
xa
x
?
 
Solution 21: 
Let f(x) = 
? ? sin
cos
xa
x
?
 
By quotient rule, 
? ?
? ? ? ?
? ?
? ? ? ? ? ?
? ?
2
2
cos sin sin cos
'
cos
cos sin sin sin
' ...
cos
dd
x x a x a x
dx dx
fx
x
d
x x a x a x
dx
f x i
x
? ? ? ??
??
?
? ? ? ? ??
??
?
 
Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) 
By first principle, 
? ?
? ?
? ? ? ?
0
0
0
0
0
()
' lim
1
lim sin sin
1
lim 2cos sin
22
1 2 2
lim 2cos sin
2
sin
22 2
lim cos
2
h
h
h
h
h
g x h g x
gx
h
x h a x a
h
x h a x a x h a x a
h
x a h h
hh
h
x a h
h h
?
?
?
?
?
??
?
? ? ? ? ? ??
??
? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
?? ??
??
??
???? ??
??
?
??
??
??
??
?
??
?
?? ?
? ? ? ?
0
0
2
0
sin
2 2 h 2
limcos .lim As h 0 0
2
2
2 2 sinh
cos 1 lim 1
2
cos ...
h h
h
h
x a h
h h
xa
h
x a ii
?
?
?
??
??
??
??
?
??
?
? ??
?? ??
??
??
?? ?? ? ? ? ?
??
? ? ? ?
??
??
??
??
? ? ? ?
??
??
??
?? ??
? ? ? ? ?
? ? ?
??
??
? ? ? ?
??
 
From (i) and (ii), we obtain 
Page 2


Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 21: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers):   
? ? sin
cos
xa
x
?
 
Solution 21: 
Let f(x) = 
? ? sin
cos
xa
x
?
 
By quotient rule, 
? ?
? ? ? ?
? ?
? ? ? ? ? ?
? ?
2
2
cos sin sin cos
'
cos
cos sin sin sin
' ...
cos
dd
x x a x a x
dx dx
fx
x
d
x x a x a x
dx
f x i
x
? ? ? ??
??
?
? ? ? ? ??
??
?
 
Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) 
By first principle, 
? ?
? ?
? ? ? ?
0
0
0
0
0
()
' lim
1
lim sin sin
1
lim 2cos sin
22
1 2 2
lim 2cos sin
2
sin
22 2
lim cos
2
h
h
h
h
h
g x h g x
gx
h
x h a x a
h
x h a x a x h a x a
h
x a h h
hh
h
x a h
h h
?
?
?
?
?
??
?
? ? ? ? ? ??
??
? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
?? ??
??
??
???? ??
??
?
??
??
??
??
?
??
?
?? ?
? ? ? ?
0
0
2
0
sin
2 2 h 2
limcos .lim As h 0 0
2
2
2 2 sinh
cos 1 lim 1
2
cos ...
h h
h
h
x a h
h h
xa
h
x a ii
?
?
?
??
??
??
??
?
??
?
? ??
?? ??
??
??
?? ?? ? ? ? ?
??
? ? ? ?
??
??
??
??
? ? ? ?
??
??
??
?? ??
? ? ? ? ?
? ? ?
??
??
? ? ? ?
??
 
From (i) and (ii), we obtain 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
? ? ? ?
? ?
2
2
2
cos .cos sin sin
'
cos
cos
cos
cos
cos
x x a x x a
fx
x
x a x
x
a
x
? ? ?
?
??
?
?
 
 
 
Question 22: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): x
4
 (5 sin x - 3 cos x) 
Solution 22: 
Let 
4
( ) (5sin 3cos ) f x x x x ?? 
By product rule, 
? ?
? ? ? ? ? ?
? ? ? ?
? ?
44
44
43
3
'( ) (5sin 3cos ) (5sin 3cos )
5 sin 3 cos (5sin 3cos )
5cos 3 sin (5sin 3cos ) 4
5 cos 3 sin 20sin 12cos
dd
f x x x x x x x
dx dx
d d d
x x x x x x
dx dx dx
x x x x x x
x x x x x x x
? ? ? ?
??
? ? ? ?
??
??
? ? ? ? ? ??
??
? ? ? ?
 
 
 
Question 23: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): (x
2
 + 1) cos x 
Solution 23: 
Let f(x) = 
? ?
2
1 cos xx ? 
By product rule, 
? ? ? ? ? ? ? ?
? ? ? ? ? ?
22
2
2
' 1 cos cos 1
1 sin cos 2
sin sin 2 cos
dd
f x x x x x
dx dx
x x x x
x x x x x
? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
Page 3


Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 21: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers):   
? ? sin
cos
xa
x
?
 
Solution 21: 
Let f(x) = 
? ? sin
cos
xa
x
?
 
By quotient rule, 
? ?
? ? ? ?
? ?
? ? ? ? ? ?
? ?
2
2
cos sin sin cos
'
cos
cos sin sin sin
' ...
cos
dd
x x a x a x
dx dx
fx
x
d
x x a x a x
dx
f x i
x
? ? ? ??
??
?
? ? ? ? ??
??
?
 
Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) 
By first principle, 
? ?
? ?
? ? ? ?
0
0
0
0
0
()
' lim
1
lim sin sin
1
lim 2cos sin
22
1 2 2
lim 2cos sin
2
sin
22 2
lim cos
2
h
h
h
h
h
g x h g x
gx
h
x h a x a
h
x h a x a x h a x a
h
x a h h
hh
h
x a h
h h
?
?
?
?
?
??
?
? ? ? ? ? ??
??
? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
?? ??
??
??
???? ??
??
?
??
??
??
??
?
??
?
?? ?
? ? ? ?
0
0
2
0
sin
2 2 h 2
limcos .lim As h 0 0
2
2
2 2 sinh
cos 1 lim 1
2
cos ...
h h
h
h
x a h
h h
xa
h
x a ii
?
?
?
??
??
??
??
?
??
?
? ??
?? ??
??
??
?? ?? ? ? ? ?
??
? ? ? ?
??
??
??
??
? ? ? ?
??
??
??
?? ??
? ? ? ? ?
? ? ?
??
??
? ? ? ?
??
 
From (i) and (ii), we obtain 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
? ? ? ?
? ?
2
2
2
cos .cos sin sin
'
cos
cos
cos
cos
cos
x x a x x a
fx
x
x a x
x
a
x
? ? ?
?
??
?
?
 
 
 
Question 22: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): x
4
 (5 sin x - 3 cos x) 
Solution 22: 
Let 
4
( ) (5sin 3cos ) f x x x x ?? 
By product rule, 
? ?
? ? ? ? ? ?
? ? ? ?
? ?
44
44
43
3
'( ) (5sin 3cos ) (5sin 3cos )
5 sin 3 cos (5sin 3cos )
5cos 3 sin (5sin 3cos ) 4
5 cos 3 sin 20sin 12cos
dd
f x x x x x x x
dx dx
d d d
x x x x x x
dx dx dx
x x x x x x
x x x x x x x
? ? ? ?
??
? ? ? ?
??
??
? ? ? ? ? ??
??
? ? ? ?
 
 
 
Question 23: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): (x
2
 + 1) cos x 
Solution 23: 
Let f(x) = 
? ?
2
1 cos xx ? 
By product rule, 
? ? ? ? ? ? ? ?
? ? ? ? ? ?
22
2
2
' 1 cos cos 1
1 sin cos 2
sin sin 2 cos
dd
f x x x x x
dx dx
x x x x
x x x x x
? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 24: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): (ax
2
 + sin x) (p + q cos x) 
Solution 24: 
Let ? ? ? ?
2
sin f x ax x ?? (p + q cos x) 
By product rule, 
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
22
2
2
' sin cos cos sin
sin sin cos 2 cos
sin sin cos 2 cos
dd
f x ax x p q x p q x ax x
dx dx
ax x q x p q x ax x
q x ax x p q x ax x
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
 
Question 25: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers):   (x + cos x) (x – tan x) 
Solution 25: 
Let f(x) = (x + cos x) (x – tan x) 
By product rule, 
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
' (x + cos x) tan tan cos
x + cos x tan tan 1 sin
x + cos x 1 tan tan 1 sin ...
Let g x tan . Accordingly, g(x+h)=tan x+h
dd
f x x x x x x x
dx dx
dd
x x x x x
dx dx
d
x x x x i
dx
x
? ? ? ? ?
??
? ? ? ? ?
??
??
??
? ? ? ? ?
??
??
?
 
By first principle, 
? ?
? ? ? ?
? ?
? ?
0
0
0
' lim
tan tan
lim
1 sin( ) sin
lim
cos cos
h
h
h
g x h g x
gx
h
x h x
h
x h x
h x h x
?
?
?
??
?
??
?
??
?
??
??
?
??
 
Page 4


Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 21: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers):   
? ? sin
cos
xa
x
?
 
Solution 21: 
Let f(x) = 
? ? sin
cos
xa
x
?
 
By quotient rule, 
? ?
? ? ? ?
? ?
? ? ? ? ? ?
? ?
2
2
cos sin sin cos
'
cos
cos sin sin sin
' ...
cos
dd
x x a x a x
dx dx
fx
x
d
x x a x a x
dx
f x i
x
? ? ? ??
??
?
? ? ? ? ??
??
?
 
Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) 
By first principle, 
? ?
? ?
? ? ? ?
0
0
0
0
0
()
' lim
1
lim sin sin
1
lim 2cos sin
22
1 2 2
lim 2cos sin
2
sin
22 2
lim cos
2
h
h
h
h
h
g x h g x
gx
h
x h a x a
h
x h a x a x h a x a
h
x a h h
hh
h
x a h
h h
?
?
?
?
?
??
?
? ? ? ? ? ??
??
? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
?? ??
??
??
???? ??
??
?
??
??
??
??
?
??
?
?? ?
? ? ? ?
0
0
2
0
sin
2 2 h 2
limcos .lim As h 0 0
2
2
2 2 sinh
cos 1 lim 1
2
cos ...
h h
h
h
x a h
h h
xa
h
x a ii
?
?
?
??
??
??
??
?
??
?
? ??
?? ??
??
??
?? ?? ? ? ? ?
??
? ? ? ?
??
??
??
??
? ? ? ?
??
??
??
?? ??
? ? ? ? ?
? ? ?
??
??
? ? ? ?
??
 
From (i) and (ii), we obtain 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
? ? ? ?
? ?
2
2
2
cos .cos sin sin
'
cos
cos
cos
cos
cos
x x a x x a
fx
x
x a x
x
a
x
? ? ?
?
??
?
?
 
 
 
Question 22: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): x
4
 (5 sin x - 3 cos x) 
Solution 22: 
Let 
4
( ) (5sin 3cos ) f x x x x ?? 
By product rule, 
? ?
? ? ? ? ? ?
? ? ? ?
? ?
44
44
43
3
'( ) (5sin 3cos ) (5sin 3cos )
5 sin 3 cos (5sin 3cos )
5cos 3 sin (5sin 3cos ) 4
5 cos 3 sin 20sin 12cos
dd
f x x x x x x x
dx dx
d d d
x x x x x x
dx dx dx
x x x x x x
x x x x x x x
? ? ? ?
??
? ? ? ?
??
??
? ? ? ? ? ??
??
? ? ? ?
 
 
 
Question 23: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): (x
2
 + 1) cos x 
Solution 23: 
Let f(x) = 
? ?
2
1 cos xx ? 
By product rule, 
? ? ? ? ? ? ? ?
? ? ? ? ? ?
22
2
2
' 1 cos cos 1
1 sin cos 2
sin sin 2 cos
dd
f x x x x x
dx dx
x x x x
x x x x x
? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 24: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): (ax
2
 + sin x) (p + q cos x) 
Solution 24: 
Let ? ? ? ?
2
sin f x ax x ?? (p + q cos x) 
By product rule, 
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
22
2
2
' sin cos cos sin
sin sin cos 2 cos
sin sin cos 2 cos
dd
f x ax x p q x p q x ax x
dx dx
ax x q x p q x ax x
q x ax x p q x ax x
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
 
Question 25: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers):   (x + cos x) (x – tan x) 
Solution 25: 
Let f(x) = (x + cos x) (x – tan x) 
By product rule, 
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
' (x + cos x) tan tan cos
x + cos x tan tan 1 sin
x + cos x 1 tan tan 1 sin ...
Let g x tan . Accordingly, g(x+h)=tan x+h
dd
f x x x x x x x
dx dx
dd
x x x x x
dx dx
d
x x x x i
dx
x
? ? ? ? ?
??
? ? ? ? ?
??
??
??
? ? ? ? ?
??
??
?
 
By first principle, 
? ?
? ? ? ?
? ?
? ?
0
0
0
' lim
tan tan
lim
1 sin( ) sin
lim
cos cos
h
h
h
g x h g x
gx
h
x h x
h
x h x
h x h x
?
?
?
??
?
??
?
??
?
??
??
?
??
 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
? ?
0
sin( )cos sin cos
1
lim
cos cos
h
x h x x x h
h x h x
?
?? ? ? ?
?
??
?
??
 
? ?
? ?
? ?
? ?
? ?
0
0
00
2
2
1 1 in( )
lim
cos cos
1 1 sinh
lim
cos cos
1 sinh 1
. lim . lim
cos cos
11
.1.
cos cos 0
1
cos
sec
h
h
hh
s x h x
x h x h
x h x h
x h x h
xx
x
x ii
?
?
??
??
??
?
??
?
??
??
?
??
?
??
??
??
?
??
??
??
?
??
??
?
?
?
?
 
 
? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
2
2
2
Therefore, from (i) and ii ,We obtain
f' x cos (1 sec ) tan 1 sin
cos tan tan 1 sin
tan cos tan 1 sin
x x x x x x
x x x x x x
x x x x x x
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
 
Question 26: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non 
zero constants and m and n are integers):   
4 5sin
3 7cos
xx
xx
?
?
 
Solution 26: 
Let f(x) = 
4 5sin
3 7cos
xx
xx
?
?
 
By quotient rule, 
? ?
? ? ? ? ? ? ? ?
? ?
2
3 7cos 4 5sin 4 5sin 3 7cos
'
3 7cos
dd
x x x x x x x x
dx dx
fx
xx
? ? ? ? ?
?
?
 
Page 5


Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 21: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers):   
? ? sin
cos
xa
x
?
 
Solution 21: 
Let f(x) = 
? ? sin
cos
xa
x
?
 
By quotient rule, 
? ?
? ? ? ?
? ?
? ? ? ? ? ?
? ?
2
2
cos sin sin cos
'
cos
cos sin sin sin
' ...
cos
dd
x x a x a x
dx dx
fx
x
d
x x a x a x
dx
f x i
x
? ? ? ??
??
?
? ? ? ? ??
??
?
 
Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) 
By first principle, 
? ?
? ?
? ? ? ?
0
0
0
0
0
()
' lim
1
lim sin sin
1
lim 2cos sin
22
1 2 2
lim 2cos sin
2
sin
22 2
lim cos
2
h
h
h
h
h
g x h g x
gx
h
x h a x a
h
x h a x a x h a x a
h
x a h h
hh
h
x a h
h h
?
?
?
?
?
??
?
? ? ? ? ? ??
??
? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
? ? ? ? ? ? ? ?
?
? ? ? ? ??
? ? ? ? ??
?? ??
??
??
???? ??
??
?
??
??
??
??
?
??
?
?? ?
? ? ? ?
0
0
2
0
sin
2 2 h 2
limcos .lim As h 0 0
2
2
2 2 sinh
cos 1 lim 1
2
cos ...
h h
h
h
x a h
h h
xa
h
x a ii
?
?
?
??
??
??
??
?
??
?
? ??
?? ??
??
??
?? ?? ? ? ? ?
??
? ? ? ?
??
??
??
??
? ? ? ?
??
??
??
?? ??
? ? ? ? ?
? ? ?
??
??
? ? ? ?
??
 
From (i) and (ii), we obtain 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
? ? ? ?
? ?
2
2
2
cos .cos sin sin
'
cos
cos
cos
cos
cos
x x a x x a
fx
x
x a x
x
a
x
? ? ?
?
??
?
?
 
 
 
Question 22: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): x
4
 (5 sin x - 3 cos x) 
Solution 22: 
Let 
4
( ) (5sin 3cos ) f x x x x ?? 
By product rule, 
? ?
? ? ? ? ? ?
? ? ? ?
? ?
44
44
43
3
'( ) (5sin 3cos ) (5sin 3cos )
5 sin 3 cos (5sin 3cos )
5cos 3 sin (5sin 3cos ) 4
5 cos 3 sin 20sin 12cos
dd
f x x x x x x x
dx dx
d d d
x x x x x x
dx dx dx
x x x x x x
x x x x x x x
? ? ? ?
??
? ? ? ?
??
??
? ? ? ? ? ??
??
? ? ? ?
 
 
 
Question 23: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): (x
2
 + 1) cos x 
Solution 23: 
Let f(x) = 
? ?
2
1 cos xx ? 
By product rule, 
? ? ? ? ? ? ? ?
? ? ? ? ? ?
22
2
2
' 1 cos cos 1
1 sin cos 2
sin sin 2 cos
dd
f x x x x x
dx dx
x x x x
x x x x x
? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
Question 24: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers): (ax
2
 + sin x) (p + q cos x) 
Solution 24: 
Let ? ? ? ?
2
sin f x ax x ?? (p + q cos x) 
By product rule, 
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
22
2
2
' sin cos cos sin
sin sin cos 2 cos
sin sin cos 2 cos
dd
f x ax x p q x p q x ax x
dx dx
ax x q x p q x ax x
q x ax x p q x ax x
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
 
Question 25: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non-zero constants and m and n are integers):   (x + cos x) (x – tan x) 
Solution 25: 
Let f(x) = (x + cos x) (x – tan x) 
By product rule, 
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
' (x + cos x) tan tan cos
x + cos x tan tan 1 sin
x + cos x 1 tan tan 1 sin ...
Let g x tan . Accordingly, g(x+h)=tan x+h
dd
f x x x x x x x
dx dx
dd
x x x x x
dx dx
d
x x x x i
dx
x
? ? ? ? ?
??
? ? ? ? ?
??
??
??
? ? ? ? ?
??
??
?
 
By first principle, 
? ?
? ? ? ?
? ?
? ?
0
0
0
' lim
tan tan
lim
1 sin( ) sin
lim
cos cos
h
h
h
g x h g x
gx
h
x h x
h
x h x
h x h x
?
?
?
??
?
??
?
??
?
??
??
?
??
 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ?
? ?
0
sin( )cos sin cos
1
lim
cos cos
h
x h x x x h
h x h x
?
?? ? ? ?
?
??
?
??
 
? ?
? ?
? ?
? ?
? ?
0
0
00
2
2
1 1 in( )
lim
cos cos
1 1 sinh
lim
cos cos
1 sinh 1
. lim . lim
cos cos
11
.1.
cos cos 0
1
cos
sec
h
h
hh
s x h x
x h x h
x h x h
x h x h
xx
x
x ii
?
?
??
??
??
?
??
?
??
??
?
??
?
??
??
??
?
??
??
??
?
??
??
?
?
?
?
 
 
? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
2
2
2
Therefore, from (i) and ii ,We obtain
f' x cos (1 sec ) tan 1 sin
cos tan tan 1 sin
tan cos tan 1 sin
x x x x x x
x x x x x x
x x x x x x
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
 
Question 26: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed non 
zero constants and m and n are integers):   
4 5sin
3 7cos
xx
xx
?
?
 
Solution 26: 
Let f(x) = 
4 5sin
3 7cos
xx
xx
?
?
 
By quotient rule, 
? ?
? ? ? ? ? ? ? ?
? ?
2
3 7cos 4 5sin 4 5sin 3 7cos
'
3 7cos
dd
x x x x x x x x
dx dx
fx
xx
? ? ? ? ?
?
?
 
Class XI Chapter 13 – Limits and Derivatives   Maths 
______________________________________________________________________________ 
 
? ? ? ? ? ? ? ?
? ?
2
3 7cos 4 5 sin 4 5sin 3 7 cos
3 7cos
d d d d
x x x x x x x x
dx dx dx dx
xx
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
?
?
 
? ? ? ? ? ? ? ?
? ?
? ?
? ?
? ?
2
2 2 2
2
22
2
3 7cos 4 5cos 4 5sin 3 7sin
3 7cos
12 15 cos 28cos 35cos 12 28 sin 15sin 35 cos sin
3 7cos
15 cos 28cos 28 sin 15sin 35(cos sin )
3 7cos
35 15 cos 28cos 28 sin 15sin
3
x x x x x x x
xx
x x x x x x x x x x x
xx
x x x x x x x x
xx
x x x x x x
? ? ? ? ?
?
?
? ? ? ? ? ? ? ?
?
?
? ? ? ? ?
?
?
? ? ? ?
?
? ?
2
7cos xx ?
 
 
 
Question 27: 
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s 
are fixed nonzero constants and m and n are integers): 
2
cos
4
sin
x
x
? ??
??
??
 
Solution 27: 
Let f(x) = 
2
cos
4
sin
x
x
? ??
??
??
 
By quotient rule, 
? ?
? ? ? ?
? ?
22
2
2
2
2
sin sin
' cos .
4 sin
sin .2 cos
cos .
4 sin
cos 2sin cos
4
sin
dd
x x x x
dx dx
fx
x
x x x x
x
x x x x
x
?
?
?
??
?
??
??
?
?? ??
??
??
??
?? ? ??
?
????
??
??
?
?
 
 
 
Read More
209 videos|443 docs|143 tests

Up next

FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

1. What is the concept of limits in calculus?
Ans. In calculus, the concept of limits refers to the value that a function approaches as the input variable gets closer and closer to a certain value. It helps us understand the behavior of functions and their values at specific points.
2. How do limits help in finding derivatives?
Ans. Limits play a crucial role in finding derivatives. The derivative of a function at a point is defined as the limit of the slope of a secant line passing through that point and a nearby point on the function. By taking the limit as the two points get closer, we can find the instantaneous rate of change, which is the derivative.
3. What is the difference between a one-sided limit and a two-sided limit?
Ans. A one-sided limit is concerned with the behavior of a function as the input approaches a certain value from either the left or the right side. It only considers the values of the function from one side of the given point. On the other hand, a two-sided limit considers the values of the function from both the left and right sides of the given point.
4. Can limits be used to determine the continuity of a function?
Ans. Yes, limits can be used to determine the continuity of a function. If the limit of a function at a certain point exists and is equal to the value of the function at that point, then the function is continuous at that point. Limits help us analyze the behavior of a function and identify any potential discontinuities.
5. How are limits useful in real-life applications?
Ans. Limits have various real-life applications. In physics, limits are used to calculate instantaneous velocity and acceleration. In economics, limits help in determining marginal cost and marginal revenue. In engineering, limits are used to analyze the stability and efficiency of systems. Overall, limits provide a mathematical framework to understand and solve problems in various fields.
209 videos|443 docs|143 tests
Download as PDF

Up next

Explore Courses for JEE exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

Objective type Questions

,

ppt

,

shortcuts and tricks

,

Summary

,

Previous Year Questions with Solutions

,

Extra Questions

,

study material

,

NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

,

Semester Notes

,

past year papers

,

NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

,

pdf

,

Free

,

video lectures

,

practice quizzes

,

MCQs

,

mock tests for examination

,

Sample Paper

,

Viva Questions

,

Exam

,

Important questions

,

NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

;