Derivatives and Inverse Derivatives

1. For the point (a,b) on the x,y plane the derivative of the inverse function derived at point b is the reciprocal of the derivative of the function at point a.

2. If F(x) is an invertible function, then for any point (a,b) on F(x):


3. The volume equation is an invertible function because its inverse is a function.

4. This is my thought process:




5.
Black is the original volume function, red is the inverse of that, blue is the derivative of the original function at x = 1, and green is the derivative of the inverse at point y = 1
The functions I used were as follows:
Y = (4/3)*pi*x^3
Y^-1 = ((3x)/(4*pi))^(1/3)
Y' = 4*pi*(x-1)+((4/3)*pi)
Y'^-1 = ((1/(4*pi))(x-((4/3)*pi)))+1

6. 1/(4*pi) = the inverse of 4*pi

7. I came to class on Wednesday because I would not have learned the same amount on my own. I needed the review that we did in class and I learn better with interaction. I also probably don't have the initiative to do the work on my own, so it is good that I came to class because otherwise I might not have studied thoroughly.