Since the problem asked you to find two numbers, I have no idea what you could mean by "I get 0".
What function are you differentiating? The problem asked for two numbers "whose sum is a minimum". The sum of two numbers, x and y, is f(x,y)= x+ y. That is what you want to differentiate. You are also told that their product is 81: xy= 81 so y= 81/x so that .
Ok,
you need to understand the point of having 2 clues.
One clue allows you to write one variable "in terms of the other".
Then you can write an equation in one variable only.
y=81/x is only one component of the sum equation.
You want the sum to be a minimum, but you must work with the entire sum formula.
xy=81
x+y=sum.
y=81/x
Therefore x+81/x=sum.
sum is a minimum, so the derivative, wrt x, of the sum is zero.
See?