Now we want to be able to use the chain rule on multi-variable functions.
Lets start with the two-variable function and then generalize from there.
Lets say that both x and y can be expressed as functions of another variable t . Now, by linear approximation we know where the above error functions go to zero as delta x and delta y go to zero. We divide by delta t and get, Now as we let delta t go to zero, delta x, delta y, and the error functions go to zero and we get the expression. This formula extends as one would expect. They can be functions of other variables, too.