How would I use implicit differentiation to differentiate functions such as: y=tan^-1(ax^2+b) in the form of dy/dx=.....?

First you must write the function in terms on something you know how to differentiate, for example... by taking tan (..) of both sides the equation becomes, tan(y)= ax² +b. We then use implicit differentiation. So in our case, tan(y) goes too sec²(y)*dy/dx when differentiating y with respect to x on the left hand side of our re-aranged equation, using the chain rule. The right hand side is completed as normal with respect to x. Leaving us with dy/dx * sec²(y) = 2ax.  This gets us to a final answer of dy/dx = 2ax / (sec²(y)) = 2ax * cos²(y). Using the identity Sin²(x)+Cos²(x)=1 we can get the result in terms of x.