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
?
?
?
??
?
??
??
?
?? ??
??
??
??
?? ? ??
?
????
??
??
?
?
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