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 to.

Inverse Sine

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.  When 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.

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,

Then,

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.

Inverse Cosine

Now let’s take a look at the inverse cosine.  Here is the definition for the inverse cosine.

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 .

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

then,

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 derivative.

So, the derivative of the inverse cosine is nearly identical to the derivative of the inverse sine.  The only difference is the negative sign.

Inverse Tangent

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.

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 below.

From this graph we can see that

The tangent and inverse tangent functions are inverse functions so,

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 inverse tangent.

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.