Two young mathematicians discuss the derivative of inverse functions.

Check out this dialogue between two calculus students (based on a true story):
Riley, I have a calculus question.
Hit me with it.
What’s the derivative of ?
Hmmm…we haven’t talked about that yet in our class.
I know! But maybe we can figure it out.
Well and now we can use the chain rule to take its derivative
But is this right?

Let’s see if we can figure out if Devyn and Riley are correct. Start by looking at a plot of :

Let . Use the plot to determine the intervals(s) where the function is increasing.

From the graph it seems that the function is increasing on the interval

On the other hand,

What is the sign of on the interval ?
positive negative
Complete the sentence below:

Since the sign of on the interval is

positive negative
the function must be
increasing on the interval decreasing on the interval

In light of the problems above, is it possible that

yes no
When our friends wrote , what do they think the “” represents? Are they correct?
Riley thinks that we can use the power rule on the , which tells us that the students are using as an exponent for the tangent function. However, in the case of inverse functions such as , the is not an exponent.