In this section we explore the relationship between the derivative of a function and the derivative of its inverse. For functions whose derivatives we already know, we can use this relationship to find derivatives of inverses without having to use the limit definition of the derivative. In particular, we will apply the formula for derivatives of inverse functions to trigonometric functions. This formula may also be used to extend the power rule to rational exponents.
We begin by considering a function and its inverse. If f(x)
is both invertible and differentiable, it seems reasonable that the inverse of f(x)
is also differentiable. [link] shows the relationship between a function f(x)
and its inverse f−1(x).
Look at the point (a,f−1(a))
on the graph of f−1(x)
having a tangent line with a slope of (f−1)′(a)=pq.
This point corresponds to a point (f−1(a),a)
on the graph of f(x)
having a tangent line with a slope of f′(f−1(a))=qp.
Thus, if f−1(x)
is differentiable at a,
then it must be the case that
We may also derive the formula for the derivative of the inverse by first recalling that x=f(f−1(x)).
Then by differentiating both sides of this equation (using the chain rule on the right), we obtain
Solving for (f−1)′(x),
we obtain
We summarize this result in the following theorem.
Let f(x)
be a function that is both invertible and differentiable. Let y=f−1(x)
be the inverse of f(x).
For all x
satisfying f′(f−1(x))≠0,
Alternatively, if y=g(x)
is the inverse of f(x),
then
Use the inverse function theorem to find the derivative of g(x)=x+2x.
Compare the resulting derivative to that obtained by differentiating the function directly.
The inverse of g(x)=x+2x
is f(x)=2x−1.
Since g′(x)=1f′(g(x)),
begin by finding f′(x).
Thus,
Finally,
We can verify that this is the correct derivative by applying the quotient rule to g(x)
to obtain
Use the inverse function theorem to find the derivative of g(x)=1x+2.
Compare the result obtained by differentiating g(x)
directly.
Use the preceding example as a guide.
Use the inverse function theorem to find the derivative of g(x)=3√x.
The function g(x)=3√x
is the inverse of the function f(x)=x3.
Since g′(x)=1f′(g(x)),
begin by finding f′(x).
Thus,
Finally,
Find the derivative of g(x)=5√x
by applying the inverse function theorem.
is the inverse of f(x)=x5.
From the previous example, we see that we can use the inverse function theorem to extend the power rule to exponents of the form 1n,
where n
is a positive integer. This extension will ultimately allow us to differentiate xq,
where q
is any rational number.
The power rule may be extended to rational exponents. That is, if n
is a positive integer, then
Also, if n
is a positive integer and m
is an arbitrary integer, then
The function g(x)=x1/n
is the inverse of the function f(x)=xn.
Since g′(x)=1f′(g(x)),
begin by finding f′(x).
Thus,
Finally,
To differentiate xm/n
we must rewrite it as (x1/n)m
and apply the chain rule. Thus,
□
Find the equation of the line tangent to the graph of y=x2/3
at x=8.
First find dydx
and evaluate it at x=8.
Since
the slope of the tangent line to the graph at
is
Substituting
into the original function, we obtain
Thus, the tangent line passes through the point
Substituting into the point-slope formula for a line, we obtain the tangent line
Find the derivative of
Use the chain rule.
We now turn our attention to finding derivatives of inverse trigonometric functions. These derivatives will prove invaluable in the study of integration later in this text. The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic functions. Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to be trigonometric functions. Here, for the first time, we see that the derivative of a function need not be of the same type as the original function.
Use the inverse function theorem to find the derivative of
Since for
in the interval
is the inverse of
begin by finding
Since
we see that
To see that
consider the following argument. Set
In this case,
where
We begin by considering the case where
Since
is an acute angle, we may construct a right triangle having acute angle
a hypotenuse of length
and the side opposite angle
having length
From the Pythagorean theorem, the side adjacent to angle
has length
This triangle is shown in [link]. Using the triangle, we see that
In the case where
we make the observation that
and hence
Now if
or
or
and since in either case
and
we have
Consequently, in all cases,
Apply the chain rule to the formula derived in [link] to find the derivative of
and use this result to find the derivative of
Applying the chain rule to
we have
Now let
so
Substituting into the previous result, we obtain
Use the inverse function theorem to find the derivative of
The derivatives of the remaining inverse trigonometric functions may also be found by using the inverse function theorem. These formulas are provided in the following theorem.
Find the derivative of
Let
so
Substituting into [link], we obtain
Simplifying, we have
Find the derivative of
By applying the product rule, we have
Find the derivative of
Use [link]. with
The position of a particle at time
is given by
for
Find the velocity of the particle at time
Begin by differentiating
in order to find
Thus,
Simplifying, we have
Thus,
Find the equation of the line tangent to the graph of
at
is the slope of the tangent line.
whenever
and
is differentiable.
For the following exercises, use the graph of
to
and
a.* * *
b.
a.* * *
b.
For the following exercises, use the functions
to find
at
and
at
a. 6, b.
c.
a.
b.
c.
For each of the following functions, find
For each of the given functions
at the indicated point
and
at the indicated point.
a.
b.
a.
b.
For the following exercises, find
for the given function.
For the following exercises, use the given values to find
[T] The position of a moving hockey puck after
seconds is
where
is in meters.
and
seconds.
a.
b.
c.
d. The hockey puck is decelerating/slowing down at 2, 4, and 6 seconds.
[T] A building that is 225 feet tall casts a shadow of various lengths
as the day goes by. An angle of elevation
is formed by lines from the top and bottom of the building to the tip of the shadow, as seen in the following figure. Find the rate of change of the angle of elevation
when
feet.
[T] A pole stands 75 feet tall. An angle
is formed when wires of various lengths of
feet are attached from the ground to the top of the pole, as shown in the following figure. Find the rate of change of the angle
when a wire of length 90 feet is attached.
radians per foot
[T] A television camera at ground level is 2000 feet away from the launching pad of a space rocket that is set to take off vertically, as seen in the following figure. The angle of elevation of the camera can be found by
where
is the height of the rocket. Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5000 feet apart.
[T] A local movie theater with a 30-foot-high screen that is 10 feet above a person’s eye level when seated has a viewing angle
(in radians) given by
where
is the distance in feet away from the movie screen that the person is sitting, as shown in the following figure.
for
and 20.
for
and 40
should the person stand to maximize his or her viewing angle?
a.
b.
c. As a person moves farther away from the screen, the viewing angle is increasing, which implies that as he or she moves farther away, his or her screen vision is widening. d.
e. As the person moves beyond 20 feet from the screen, the viewing angle is decreasing. The optimal distance the person should stand for maximizing the viewing angle is 20 feet.
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