Suppose $f$ is injective and $f(1)=5$ and $f'(1)=3$. Then a quick application of the inverse function theorem gives $(f^{-1})'(5)=1/3$. However, what is incorrect in the following?:
$$(f^{-1})'(5)=(f^{-1}(5))'=f'(1)=3$$
My question is, how do I explain to a student that the first equality above is incorrect?
The error is in the step
$$(f^{-1})'(5) = (f^{-1}(5))'$$
Let $f^{-1} = g$
$$\implies g'(5) = (g(5))'$$
What you have on the left hand side is the value of the derivative of the inverse evaluated at $x=5$. What you have on the right doesn't make sense - you are taking the derivative of an instance of the function at a point - which makes no sense, as you can only define a derivative of a function
I'm not sure what grade this is being discussed in, but if derivatives have been introduced I'm assuming there is an understanding of the basics of limits and differentiability