You say goodbye to your friend at the intersection of two perpendicular roads. At time [math] you drive off North at a (constant) speed [math] and your friend drives West at a (constant) speed [math]. You badly want to know: how fast is the distance between you and your friend increasing at time [math]?

Enter here the derivative with respect to [math] of the distance between you and your friend: Note: the next part will be MUCH easier if you simplify your answer to this part as much as possible.

Being scientifically minded you ask yourself how does the speed of separation change with time. In other words, what is the second derivative of the distance between you and your friend?

Suppose that after your friend takes off (at time [math]) you linger for an hour to contemplate the spot on which he or she was standing. After that hour you drive off too (to the North). How fast is the distance between you and your friend increasing at time [math] (greater than one hour)?

Again, you ask what is the second derivative of your separation:

Notice how lingering can make things harder, mathematically speaking.