so dy is an approximation of deltay, they are not always exactly the same
Dan's formula looks neater than mine, so let me use that.
let delta x = a, then we have
delta y = f(x + a) - f(x), why? here's an explanation. it would be better with a graph, but let's see if i can do it in words.
delta y is the change in y that results from a change in x. delta x = a is a small change in x.
now remember that f(x) gives the y value for any x
so if the first x coordinate is x and then we have a small change of a, so that the second coordinate is x + a. then the change in y will be given by the difference of the y values between the two. the y value at x is f(x) and the y value at x + a is f(x + a) so the change in y will be the difference between the two y values, namely f(x + a) - f(x)
see the quotient rule makes it complicated. what i did was to use the laws of logarithms to simplify. the laws i used are as follows:
ln(x)^n = nln(x) ..........this is how we got to take the 1/2 power down in front.
ln(x/y) = ln(x) - ln(y) ..........i used this to separate the fraction into two managable pieces.