Let 𝑔 be the inverse of 𝑓. Using the values in the table, find 𝑔 prime of one. And we’ve been given a table of values that tells us when 𝑥 is one, the value of 𝑓 of 𝑥 is negative five and the value of the inverse of 𝑓 𝑔 of 𝑥 is two and when 𝑥 is one, the first derivative of 𝑓 is negative two. Similarly, when 𝑥 is two we have that 𝑓 of 𝑥 is negative nine, 𝑔 of 𝑥 is four, and the first derivative of 𝑓 at 𝑥 equals two is one.
So we’re looking for 𝑔 prime of one where 𝑔 is the inverse function of 𝑓. So we can use the formula for derivatives of inverse functions. That is, for a function 𝑓 with inverse 𝑔, we have that 𝑔 prime of 𝑦 equals one over 𝑓 prime of 𝑔 of 𝑦. And this is valid, provided that the denominator is not equal to zero. So because we’re trying to find 𝑔 prime of one, let’s replace 𝑦 with one in the formula. So for this composite function on the denominator 𝑓 prime of 𝑔 of one, let’s work from the inside out.
So let’s, firstly, find the value of 𝑔 of one. We can find this information in the table. When 𝑥 is one, 𝑔 of 𝑥 is two. So 𝑔 of one is two, which means that we now need to find one over 𝑓 prime of two. So let’s use the table we’re given in the question. When 𝑥 is two, we have that 𝑓 prime of 𝑥 is one. So 𝑓 prime of two is one, which gives us that 𝑔 prime of one is equal to one over one, which is of course one.