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Section 5.3 Section 5.3 2014320143
Derivatives of Inverse FunctionsDerivatives of Inverse Functions
Objective: To find the derivative Objective: To find the derivative of the inverse of a function.of the inverse of a function.
Inverse functions are basically functions that “cancel” when we perform a composition.
Formally, we say that the function g (x) is the inverse function of f (x) if:
Inverse Functions - ReviewInverse Functions - Review
gxxxgf ofdomain in the allfor )( and
fxxxfg ofdomain in the allfor )(
The inverse function exists (without any constraints) if it satisfies one condition:
1. A function has an inverse iff it is one-to-one For every x, there is only one y and for
every y, there is only one x. If f is strictly monotonic for its entire
domain, then it is one-to-one and therefore has an inverse function.
Inverse FunctionsInverse Functions The graph of the inverse of
f (x), f -1(x), is the reflection of f (x) across the line y = x.
Example 1 – Example 1 – Verifying Inverse Verifying Inverse FunctionsFunctions
Show that the functions are inverse functions of each other.
and
Solution: Because the domains and ranges of both f and g consist of all real numbers, you can conclude that both composite functions exist for all x.The compositions f (g(x)) and g(f(x)) are given by:
Because f (g(x)) = x and g(f (x)) = x, you can
conclude that f and g are inverse functions of
each other (see Figure 5.11).
Figure 5.11
Example 1 – Example 1 – SolutionSolutioncont'd
The idea of a reflection of the graph of f in the line y = x is generalized in the following theorem.
Figure 5.12
Inverse FunctionsInverse Functions
Derivative of an Inverse FunctionDerivative of an Inverse Function
116 3 3
3f f
6Find f
16 3Find f and f
If (x,y) is a point on the graph of a function, the derivative of the function at x will be the reciprocal of the derivative of its inverse at y.
Graphs of inverse functions have reciprocal slopes at inverse points.
16 3, 3 6f f
Derivative of an Inverse FunctionDerivative of an Inverse Function
If (x,y) is a point on the graph of a function, the derivative of the function at x will be the reciprocal of the derivative of its inverse at y.
Graphs of inverse functions have reciprocal slopes at inverse points.
x
y
x
y
The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of
what this formula is actually stating:
0)(,)(
1)(
xgf
xgfxg
Let this function be f (x)
The inverse function will be reflected across the line y = x
f (x)
g (x)
Let this function be f -1 (x) = g (x)
x
y
The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of
what this formula is actually stating:
0)(,)(
1)(
xgf
xgfxg
f (x)
g (x)
Let’s say we wish to find the slope of the tangent line of g (x) at x = 5
5
The coordinate will be (5, g (5))
(5, g (5))
The slope of the tangent line will be some constant; we’ll call it g ‘(x)
x
y
The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of
what this formula is actually stating:
0)(,)(
1)(
xgf
xgfxg
f (x)
g (x)
Let’s say we wish to find the slope of the tangent line of g (x) at x = 5
5
We know that if there exists a point on g (x), then there is an inverse point on f (x)
(5, g (5))
This coordinate will be the reverse or the inverse (g (5), 5)
(g (5), 5)
x
y
The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of
what this formula is actually stating:
0)(,)(
1)(
xgf
xgfxg
f (x)
g (x)
5
According to this formula, the reciprocal of the slope of the tangent line of f(x) at x = g(5) is the SAMESAME as the slope of the tangent line of g(x) at x = 5
(5, g (5))
(g (5), 5)
In other words, if we want to find the slope of the tangent line of g(x) for some x, all we have to do is find the reciprocal of the derivative of the f(x) when f(x) = (the given x of the inverse)
The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of
what this formula is actually stating:
0)(,)(
1)(
xgf
xgfxg
x
yf (x)
g (x)
a
(a,b)
(b, a)
If we let the coordinate be more generic such as (a, b), then we could say:
0,1
)(
bfbf
ag
Graphs of inverse functions have reciprocal slopes at inverse points.
What are your options for answering this question? 1) Find the inverse, then find its derivative at x=8.
2) Find where f(x) = 8, find the derivative there, and reciprocate.
3) Use the formula.
ExamplesExamples
31 )( if )8( Find.1 xxff
First, let’s try the first option. Find the inverse by switching x and y and solving for y:
3/23
1
x
ExamplesExamples
yx 3/13yx 3/11 )( xxf
12
1 3/21
3
1
xf 43
1
3/21
83
18
f
Find the derivative of f -1:
31 )( if )8( Find.1 xxff
Now, let’s try the second option. Find where f(x) = 8, find the derivative there, and reciprocate.
f ( x ) h a s t h e p o i n t ( ? ,8 )
ExamplesExamples
31 )( if )8( Find.1 xxff
1f (x ) h a s th e p o in t (8 ,? )
( 2 ) 1 2f 1 1(8)
12f
This is usually the easiest method.
ExamplesExamples
23)( xxf
31 )( if )8( Find.1 xxff
3)( xxf 2
1
)(3
1)(
xgxf
12
1
2)8()8()8( 3/11 fg
43
1
223
1
We still need to find g (x); but since g (x) is the inverse of f (x), we know that the x and y values switch or (x, g (x)) → (g (x), x)
Now, let’s try to find the derivative of the inverse by using the formula:
0)(,)(
1)(
xgf
xgfxg
2
1
)8(3
1)8(
gf
ExamplesExamples
12)( if )2( Find.2 31 xxxff
1 1(2)
5f
1 has the point 2,?f x has the point ?, 2f x
has the point 1, 2f x
1 = 5,Since f
In Example 5, note that at the point (2, 3) the slope of the graph of f is 4 and at the point (3, 2) the slope of the graph of f
–1 is (see Figure 5.17).
Derivative of an Inverse Derivative of an Inverse FunctionFunction
HomeworkHomework Inverse Functions
Day 1: P. 347: 1, 3, 9, 11, 71-89 odd
Day 2: Derivatives of Inverses W/S