## A question on function equality

Hello to every one!
I have a question that came up when I was talking with a fellow mathematician.
I used to say that two functions are equal when the have the same formula and the same domain and codomain.
We read in a book though that two functions are equal when they have the same domain and when the values of the function are equal for the same X.
For example
$
f(x)=x^2
$
and $g(x)=x^3$ are equal when their domain is only the points 0 and 1,because f(0)=g(0)=0 and f(1)=g(1) even though their formula is different.
I thought that this definition of equality is incomplete because by saying that f(x)=g(x) then
$
\frac{df}{dx}$
= $\frac{dg}{dx}
$
but on point 1 $\frac{df}{dx}$=2 and $\frac{dg}{dx}$=3. Thus we derive two different results from to equal quantities. Therefore two functions in order to be equal should also have the same formula.
Can you please give any insight on this?