 Let's take a look at three examples of how to find derivatives of algebraic inverse functions. In this first example, we are given the function f of x equal to square root of x that contains the point 9 comma 3 and we are asked to find the derivative of the inverse of that function at the x value of 3. Now one thing you need to remember is that this point 9 comma 3 this is the point on the original f of x function. The point on the inverse then would be the ordered pair 3 comma 9 because you of course have to flip-flop the x and the y coordinates from the original function. Remember that the basic rule goes is that to find the derivative of an inverse we need one over the first derivative of the original. So let's go ahead and find the first derivative of this original function. That simply is going to be one half x to the negative one half, which I'll go ahead and rewrite as one over two square root of x. The derivative of the inverse then at 3 is going to be equal to 1 over f prime of 9. Now remember what we've talked about this 3 is the correct x value that goes with the inverse function whereas this 9 is the correct x value that goes with the original function f. So we would have one over. Now we need to substitute 9 in place of the x in the derivative. That gives us a derivative value of one-sixth So we have one over one-sixth, which of course is 6. In our second example, we are given the function f of x equal x squared plus 2 and we are asked to find the derivative of the inverse at 3. Notice we are not given an ordered pair with which to work. This 3 is the x value of the inverse, but it would have been the y value of the original. We know in general the rule for the derivative of the inverse at 3 is going to be 1 over the first derivative of the original function at its appropriate x value. Well, that's what we don't know because all we know is the 3 is the y value for the original. So the first thing we need to do is we need to find the x of that original function that goes with the y value of 3. We can do that by setting 3 equal to the function itself. We get then that x squared equals 1. x, therefore, is equal to 1. We do not use the negative 1 when we took that square root because you'll notice that the derivative function is not defined for an x value of negative 1 if this inverse was to be a function. So we will go with the primary root in this case. To find then the derivative of the inverse at 3, we need to do 1 over the first derivative of the original at 1. Going back to our original function, we find that the derivative of that simply is 2x. So when we substitute 1 into that x value in the derivative, we get in the end that our answer is 1 half. In our last example we're dealing with the function f of x equals 2x minus x cubed and we are asked to find the derivative of the inverse at negative 4. Once again, we do not have an ordered pair with which to work. We know this negative 4 is the x value obviously of the inverse, which makes it the y value off of the original. Once again, we need to find the x value that goes with that original function for which the y value would be negative 4. We can do that by setting the original function that we're given equal to negative 4. This is one that we would have to solve graphically, so feel free to do so on your graphing calculators. I would recommend inputting 2x minus x cubed as y1 and putting negative 4 under y2 and simply finding the point of intersection and you find that x equals 2. Therefore, to find the derivative of the inverse at negative 4, we need to do 1 over the derivative of the original at 2. Going back to our original function, we find the derivative of that simply is 2 minus 3x squared. If we substitute 2 into the x in that derivative function, we get negative 10. Therefore, our answer for this is 1 over negative 10.