unit 1 differential calculus - IGNOU

dy
x
Therefore, = y log e a = a loge
a
dx
In particular if y = e x , then
Inverse Function
Let y = sin – 1 x
Then sin y = x
dy
=
dx
x
e
Differentiating both sides w. r. t. x, we have
i.e.
dy
cos y = 1
dx
dy
dx
=
=
=
=
1
cos
y
1
2
cos
1 −
1
1
1 − x
y
2
sin
2
−1 π π
As we know that the range of sin x is − to , i.e. y lies between
2 2
π
to
2 2
π
− , so cos y is positive.
Similarly,
Example 1.28
Let
Solution
y =
dy
Find .
dx
= −
dx
cos 1 −
y = cos
y = tan
y = cot
−1
−1
−1
x ⇒
x ⇒
x ⇒
dy
dx
dy
dx
dy
dx
π π
(tan x)
, − < x <
4 4
y
= −
d
(tan x)
= −
x)
dx
1
=
1 + x
1
1 − x
2
1
= −
1 + x
dy 1
1
2
= −
( 1
− tan
1
2
cos x cos 2x
( 1
2
2
sec
2
− tan x)
x
Differential Calculus
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