Derivative of an Inverse Function

The property is simply the reciprocal of the derivative of the inverse of the original function. In layperson terms, it is simply one divided by the derivative of the inverse function. Here it is in symbols to hopefully make it clearer.

This is the inverse of the volume function:

This is the inverse of the volume function:

Below is the picture and description of the following functions.

(1) The inverse volume function r(V);

(2) The volume function, V(r);

(3) x=1 on V(r) and graph the tangent line through at this point.

(4) The tangent line through the “mirror” point on the graph of the inverse volume function.

(1) The inverse volume function r(V);

(2) The volume function, V(r);

(3) x=1 on V(r) and graph the tangent line through at this point.

(4) The tangent line through the “mirror” point on the graph of the inverse volume function.

The first function is the inverse volume function which is the gray function in the image of the graph. The second function is the original volume function which is the red function in the image of the graph. The third function is the tangent line to the volume equation at the point x=1. The forth function is the tangent line to the inverse volume function at the mirror point.

To find the function of the tangent line to point x=1 for the volume function you can simply type in the volume function into your calculator and use nDeriv(y1, x, x) to find the derivative at that point. Use the point slope form of a linear equation to form the tangent line at the point x=1 for the volume function. Use the derivative at the point x=1 for the volume function as the slope of the tangent line. Then insert the values of the volume function for x and y when x=1 into the point slope form as x and y. Aha, now you have the slope of the tangent line at the point x=1 for the volume function!

To find the function of the tangent line to point x=1 for the volume function you can simply type in the volume function into your calculator and use nDeriv(y1, x, x) to find the derivative at that point. Use the point slope form of a linear equation to form the tangent line at the point x=1 for the volume function. Use the derivative at the point x=1 for the volume function as the slope of the tangent line. Then insert the values of the volume function for x and y when x=1 into the point slope form as x and y. Aha, now you have the slope of the tangent line at the point x=1 for the volume function!

The function of the tangent line of the mirror point for the inverse volume function is found similarily to the previous tangent line. Because we know that this function is the inverse to the volume function the slope of the tangent line through the mirror point will be the inverse of the slope of the linear function of the tangent line to the volume function when x=1. Also we know that the x and y value at the point x=1 will be switched around for the tangent line to the inverse volume function at the mirror point. This is all that you have to do to find the function of the tangent line at the mirror point while utalizing point slope form.

I choose not to go to class and to take the virtual class because I wanted to try something new. Honestly I am quite biased and believe that you learn much better in the classroom instead of on your own by using the internet. I wanted to see if the classroom actually did have a big advantage.

I choose not to go to class and to take the virtual class because I wanted to try something new. Honestly I am quite biased and believe that you learn much better in the classroom instead of on your own by using the internet. I wanted to see if the classroom actually did have a big advantage.

I liked the descriptions of the processes you were doing. The equations themselves were a bit confusing, and could probably been simplified a bit. And the gross decimals could have been fractions which would have cleaned it up a bit. I agree that the classroom is better for learning, and I commend you for trying something new.

ReplyDeleteI think that what flying slug calls the "gross decimals" are just a nice way of being detailed. In my opinion this is wonderful, I think that this is not only a post but pretty much a whole lesson wrapped into it. Your details on how to find the different functions are a bit wordy in length but other than that I really enjoyed reading your post. I have one question though-- your equation for the inverse of a function looks odd to me, I feel like maybe you didn't complete the inverse by solving for V(r).

ReplyDeleteThank you for your wonderful post!