Suppose that y = f(x) = x^2 - 4x + 4.

Question

Then on any interval where the inverse function y = f-1(x) exists, thederivative of y = f-1(x) with respect to x is:
(Hint: x^2 - 4x + 4 can be factored and rewritten as "something" squared.)

Work thus far:
Okay, so I know that I can factor y to equal y = (x - 2)^2
Next, I can solve for the inverse function:
x = (y - 2)^2
sqrt(x) = y - 2
y = sqrt(x) + 2
Then, I can find the derivative:
y' = (1/2)x^(-1/2)

Now, my concern is that the question also hints to an implicit solution to this problem. Working out the problem implicitly, I get:
x = y^2 - 4x + 4
1 = 2y(dy/dx) - 4(dy/dx)
dy/dx = 1 / (2y - 4) where x and y satisfy the equation y = x^2 - 4x + 4.
Would this answer be correct, as well? Or should it be "where x and y satisfy the equation x = y^2 - 4y +4"?

Answer 1

from y = f(x) = x^2 - 4x + 4
forming the inverse consists of interchanging the x and y variables, so the inverse is

x = y^2 - 4y + 4

(you had -4x instead of -4y, but apparently worked it as -4y )

so 1 = 2y dy/dx - 4dy/dx
dy/dx(2y-4) = 1
dy/dx of the inverse is 1/(2y-4)

but your restriction would be y ≠ 2

Answer 2

^thats wrong. I got it wrong. sorry.

Answer 3

The answer is 1/2x^-1/2