How to calculate approximate changes in a nonlinear function first off I've not really studied math so I'm kind of in deep water.
Lets assume that $f(a,b)=y=1+a/b$ and that both variables can increase and decrease in a given scenario.
Then lets assume that the above function is used to calculate $y$ in two different time periods; $0,1$ then we can do the following:
$$y_0=1+a_0/b_0$$
$$y_1=1+a_1/b_1$$
and thereby calculate the delta function as $Δy=y_1-y_0$ or
$$Δy=(1+a_1/b_1)-(1+a_0/b_0)$$
My question is then the following, how do i compute the approximate change in the value of $Δy$ due to a change in $Δa$ and $Δb$

You can write the variation of $y$ vs. $(a,b)$ as:
$$\Delta y=|\dfrac{\partial y}{\partial a}|\Delta a+|\dfrac{\partial y}{\partial b}|\Delta b $$ so you have:
$$\Delta y=|\dfrac{1}{b}|\Delta a+|\dfrac{a}{b^2}|\Delta b$$
