In this section we are going to look at the derivatives of
the inverse trig functions. In order to
derive the derivatives of inverse trig functions we’ll need the formula from
the last section relating the derivatives of inverse functions. If f(x)
and g(x) are inverse functions then,
Recall as well that two functions are inverses if and .
We’ll go through inverse sine, inverse cosine and inverse
tangent in detail here and leave the other three to you to derive if you’d like
Let’s start with inverse sine. Here is the definition of the inverse sine.
So, evaluating an inverse trig function is the same as
asking what angle (i.e. y) did we
plug into the sine function to get x. The restrictions on y given above are there to make sure that we get a consistent
answer out of the inverse sine. We know
that there are in fact an infinite number of angles that will work and we want
a consistent value when we work with inverse sine. Using the range of angles above gives
all possible values of the sine function exactly once. If you’re not sure of that sketch out a unit
circle and you’ll see that that range of angles (the y’s) will cover all possible values of sine.
Note as well that since we also have .
Let’s work a quick example.
Example 1 Evaluate
So we are really asking what angle y solves the following equation.
and we are restricted to the values of y above.
From a unit circle we can quickly see that .
We have the following relationship between the inverse sine
function and the sine function.
In other words they are inverses of each other. This means that we can use the fact above to
find the derivative of inverse sine.
Let’s start with,
This is not a very useful formula. Let’s see if we can get a better
formula. Let’s start by recalling the
definition of the inverse sine function.
Using the first part of this definition the denominator in
the derivative becomes,
Now, recall that
Using this, the denominator is now,
Now, use the second part of the definition of the inverse
sine function. The denominator is then,
Putting all of this together gives the following derivative.
Now let’s take a look at the inverse cosine. Here is the definition for the inverse
As with the inverse since we’ve got a restriction on the
angles, y, that we get out of the
inverse cosine function. Again, if you’d like to verify this a quick sketch of
a unit circle should convince you that this range will cover all possible
values of cosine exactly once. Also, we
also have because .
Example 2 Evaluate
As with the inverse sine we are really just asking the
where y must
meet the requirements given above.
From a unit circle we can see that we must have .
The inverse cosine and cosine functions are also
inverses of each other and so we have,
To find the derivative we’ll do the same kind of work that
we did with the inverse sine above. If
we start with
Simplifying the denominator here is almost identical to the
work we did for the inverse sine and so isn’t shown here. Upon simplifying we get the following
So, the derivative of the inverse cosine is nearly identical
to the derivative of the inverse sine.
The only difference is the negative sign.
Here is the definition of the inverse tangent.
Again, we have a restriction on y, but notice that we can’t let y
be either of the two endpoints in the restriction above since tangent isn’t
even defined at those two points. To
convince yourself that this range will cover all possible values of tangent do
a quick sketch of the tangent
function and we can see that in this range we do indeed cover all possible
values of tangent. Also, in this case
there are no restrictions on x
because tangent can take on all possible values.
Example 3 Evaluate
Here we are asking,
satisfies the restrictions given above.
From a unit circle we can see that .
Because there is no restriction on x we can ask for the limits of the inverse tangent function as x goes to plus or minus infinity. To do this we’ll need the graph of the
inverse tangent function. This is shown
From this graph we can see that
The tangent and inverse tangent functions are inverse
Therefore to find the derivative of the inverse tangent
function we can start with
We then have,
Simplifying the denominator is similar to the inverse sine,
but different enough to warrant showing the details. We’ll start with the definition of the
The denominator is then,
Now, if we start with the fact that
and divide every term by cos2 y we will get,
The denominator is then,
Finally using the second portion of the definition of the
inverse tangent function gives us,
The derivative of the inverse tangent is then,
There are three more inverse trig functions but the three
shown here the most common ones.
Formulas for the remaining three could be derived by a similar process
as we did those above. Here are the derivatives
of all six inverse trig functions.
We should probably now do a couple of quick derivatives here
before moving on to the next section.
Example 4 Differentiate
the following functions.
(a) Not much to
do with this one other than differentiate each term.
forget to convert the radical to fractional exponents before using the
There is some alternate notation that is used on occasion to
denote the inverse trig functions. This