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Junior Inter Maths - 1(B)
Differentiation
t
e
Differentiation in Calculus is the first major operation on real valued functions.
Methods of differentiation include addition, subtraction, multiplication and division
along with logarithmic differentiation, implicit differentiation etc...
n
.
a
h
b
i
Note the following (éÀçCN í∫´’-Eç-îªçúÕ):
1.
d
 [f(x). g(x)]
dx
[
f'(x) g'(x)
= f(x). g(x)  + 
f(x)
g(x)
t
a
r
p
u
d
]
a
n
d
 [(ax + b) (cx + d)]
dx
e
e
.
[
w
w
a
c
= (ax + b) (cx + d)  + 
ax + b
cx + d
2.
w[
d

dx
d

dx
3.
[
f(x)

g(x)
]
ax + b

cx + d
f(x)
= 
g(x)
]
[
f'(x) g'(x)
 − 
f(x)
g(x)
ax + b
= 
cx + d
[
m cos mx
= sin mx cos nx  −
sin mx
d
 (sinmx . cosnx)
dx
t
e
n
a
c
 − 
ax + b
cx + d
d
 (sin mx cos nx)
dx
(
]
]
]
.
a
h
b
i
t
u
d
a
a
r
p )
n sin nx

cos nx
n
e
e
.
ww
= sinmx . cosnx (m cot x − n tan x)
4.
d
 [f(x). g(x). h(x)]
dx
w
[
f'(x)
g'(x)
h'(x)
= f(x). g(x). h(x)  +  + 
f(x)
g(x)
h(x)
R-29-12-14
]
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d

dx
[ x. ex. logx]
(
= x.ex. logx
= x.ex. logx
5.
(
1/
1
ex
x
 + 
+ 
x
x
e
logx
1
1
 + 1 + 
x
x log x
)
)
t
e
n
.
a
d
 (sin x. sin 2x. sin 3x ) =
dx
h
b
i
t)
a
r
(
cosx
2cos2x
3cos3x
sinx. sin 2x. sin 3x  +  + 
sinx
sin 2x
sin 3x
p
u
d
= sinx . sin2x . sin3x (cotx + 2cot2x + 3cot3x)
d
 (cos x. cos 2x. cos 3x) =
dx
a
n
e
e
.
w
w
(
)
sin x
2 sin 2x
3 sin 3x
= cos x . cos 2x . cos 3x −  −  − 
cos x
cos 2x
cos 3x
t
e
n
= − cos x cos 2x cos 3x (tanx + 2 tan 2x + 3 tan 3x)
w
6.
1 − cos 2x

1 + cos 2x
(
d

dx
)
d
=  (tan2x)
dx
b
i
t
= 2 tan x. sec2 x
(
d

dx
1 + cos 2x

1 − cos 2x
)
.
a
h
a
r
p
d
=  (cot2x)
dx
= − 2 cot x. cosec2 x
7.
d
 [sin−1 (cos x)]
dx
n
e
[ e[ (
.
w
w
u
d
a
π
d
=  sin−1 sin 
2 −x
dx
)]]
= −1
d
 [cos−1(sin x)]
dx
w
[ [ (
π
d
=  cos−1 cos 
2 −x
dx
)]] = −1
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8.

d
 sin−1 2x √1 − x2
dx
[
(
)]
d
2
=  (2 sin−1x ) = 

dx
√1 − x2
t
e
n
.
a
d
 sin−1 (3x − 4x3 )
dx
[
]
h
b
i
d
3
=  (3 sin−1x) = 

dx
√1 − x2
9.
t
a
r
d
d
−2
 [cos−1 (2x2 − 1)] =  (2cos−1x) = 

dx
dx
√1 − x2
d
 [cos−1 (4x3 − 3x)]
dx
p
u
d
a
n
d
−3
=  (3cos−1x) = 

dx
√1 − x2
e
e
.
[ww ( ) ]
d
2x
10.  sin−1 
dx
1 + x2
w
d
1 − x2
=  cos−1 2
dx
1+x
d
= 
dx
[ (
(2 tan−1x)
)]
[ (
d
2x
=  tan−1 
dx
1 − x2
2
= 2
1+x
)]
.
a
h
b
i
t
a
[ ( )]
r
p
[
] du
a
[ ( e)n
]
e
.
[w
]
w
w[ ( )]
d
a−x
11.  tan−1 
dx
1 + ax
d
= 
dx
tan−1a − tan−1x
−1
= 2
1+x
d
a+x
 tan−1 
dx
1 − ax
d
=  tan−1a + tan−1x
dx
t
e
n
1
= 2
1+x
d
3x − x3
−
1
12.  tan
2
dx
1 − 3x
d
3
=  (3 tan−1 x) = 
dx
1 + x2
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[ (
d
4x − 4x3
 tan−1 
dx
1 − 6x2 + x4
d
= 
dx
(4 tan−1 x) =
)]
t
e
4

1 + x2
n
.
a
d [f(x)]
f'(x)
13.  = 
d [g(x)]
g'(x)
h
b
i
d (ex)
ex


= 2 √ x ex.
 = 
1
d (√ x )


2√x
t
a
r
p
u
d
d (esin x)
esin x. cos x
14.  =  = esin x
d (sin x)
cos x
a
n
1


d (sin−1x)
√ 1 − x2
 =  = −1
d (cos−1 x)
−1


√ 1 − x2
e
e
.
w
w
t
e
n
w
1

d [logax]
x . log a
1
15. 
=

=

d (ax)
ax. log a
x. ax . (log a)2
[ ( )]
[ ( )]
2x
d tan−1 2
1−x
 (−1 < x < 1)
2x
d sin−1 
1 + x2
d [2 tan−1 x]
=  = 1
d [2 tan−1 x ]
.
a
h
b
i
t
a
r
p
n
e
e
.
ww
u
d
a
Writer: C. Sadasivasastry
w
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