1. Derivative of
Recall
The Derivative Rule for Inverses
If is differentiable at every point of an interval and
df
dx
is
never zero on , then 1
f −
is differentiable at every point of the
interval . The value of
1
df
dx
−
at any particular point ( )f a is
the reciprocal of the value of
df
dx
at a.
∴ =
1
We know that = ⇔ log =
Example 2 = 8 ⇔ log 8 = log 2 = 3log 2 = 3
2. Let = ln = log#
We find the inverse of
= ln = log#
= ln = log#
⇒ %&
= ∵ =
⇔ log =
Inter change x and y
= %
∴ = %
∴ The inverse of = ln is = %
3. Let us find the Derivative of at = (
Let ) = * +
=
And observe that * (
= ,- (
= ( ,- = (
We know that ∴ .
/*0+
/
1
* 2
=
+
.
/*
/
1
32
4 /
/
5
(
=
/* +
/ ( *6 (7
=
+
/*
/ (
=
+
/
/
,-
3 (
=
+
.
+
1
3 (
=
+
+/ ( = (
∴ 4 /
/
5
(
= (
Here t is arbitrary
Derivative of
∴
/
/
=