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Title: Find the differentiation or derivative of an inverse function
Description: Basic concept on how to differentiate the inverse of a function with solved examples. Proof of the formula for finding the derivative of an inverse function.
Description: Basic concept on how to differentiate the inverse of a function with solved examples. Proof of the formula for finding the derivative of an inverse function.
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Find the derivatives of inverse functions
...
f ( x) y x f
1
( y)
...
If f
inverse of f (x ) , then f 1 ( x) is differentiable at any x and
f
1
( x)
where
f
1
1
( x) denotes the
1
f [ f
1
( x)]
( x) denotes the derivative of inverse of f (x ) , f (x ) is the derivative of
f (x ) and f f
1
( x) 0
...
Differentiating f [ f 1 ( x)] x applying chain rule:
df [ f 1 ( x)] dx
dx
dx
df [ f 1 ( x)]
df [ f 1 ( x)] df 1 ( x)
df 1 ( x)
f [ f 1 ( x)] and
1 , but
[ f 1 ( x)]
dx
dx
df 1 ( x)
df 1 ( x)
Thus f [ f 1 ( x)] [ f 1 ( x)] 1
Dividing equation (1) by f [ f
[ f 1 ( x)]
1
f [ f 1 ( x)]
1
equation (1)
( x)] ,
, where [ f 1 ( x)] is the derivative of the inverse of f (x ) and
f [ f 1 ( x)] 0
...
[ f 1 ( x)] can be represented as [ f 1 ]( x)
...
If f ( x) x 3 2 x 3 , find the derivative of the inverse of f (x ) denoted by,
[ f 1 ]( x) when x 1
...
[ f 1 ( x)]
1
and
f [ f ( x)]
1
f ( x) 3x 2 2
...
f ( x) y x f
1
Applying the power rule,
( y) and the inverse of an order pair (x, y) is
(y, x)
...
Solving f ( x) 1 , we have x 3 2 x 3 1
x 3 2 x 4 0
...
for x,
(1) 3 2(1) 4 0 , (1) 3 2(1) 4 0 ,
0 3 2(0) 4 0 ,
(2) 3 (2) 4 0 ,
(2) 3 (2) 4 0 , x 2
...
[ f 1 (1)]
1
f [ f (1)]
1
, but f 1 (1) 2
...
[ f 1 (1)]
1
1
1
...
Find [ f
1
...
[ f 1 ( x)]
1
f [ f 1 ( x)]
f ( x) sin x
...
and
f ( x) y x f
1
( y) and the inverse of an order pair (x, y) is
(y, x)
...
1
2
f 1 ( 1 ) cos 1 1
2
2
, but f 1 ( 1 )
2
3
3
...
3
Example 3
...
Solution
...
f [h(2)]
f ( x) x 4 x 3
...
f ( x) y x f 1 ( y)
...
It implies the inverse of the ordered pair (2 ,y) is (y , 2)
...
Substituting the values 0 , 1 , 2 , 3
...
[ f 1 (2)]
1
f [ f (2)]
1
1
, but f
f [h(2)]
1
(2) 1 h(2)
...
h (2)
1
1
...
7
1
f (1)
Title: Find the differentiation or derivative of an inverse function
Description: Basic concept on how to differentiate the inverse of a function with solved examples. Proof of the formula for finding the derivative of an inverse function.
Description: Basic concept on how to differentiate the inverse of a function with solved examples. Proof of the formula for finding the derivative of an inverse function.