Your reasoning and your work are correct.
Hey, I just wanted to see if anyone here has a moment to check that I didn't take any unreasonable steps in the solution to this problem:
The point P moves so that at time t it is at the intersection of the curves
x*y = t
y = t*x^2
How fast is the distance of P from the origin changing at t = 2?
What I did was first to use the equation to eliminate t, which lead to x^3 = 1; i.e. P moves along x = 1. Then I substituted that back into the original equations, which in turn lead to y = t.
Using the Pythagorean theorem I then calculated P's distance d from the origin as
d^2 = 1 + y^2
Calculating the derivative of this at t = y = 2 I got 2/sqrt(5)
This is the correct answer according to the key; but I'm kinda green at this and I'm currently locked up at home with a cold, so basically I'd just be very grateful if anyone could check to see if the reasoning is correct!