Paula,
Implicit differentiation is a fairly simple process with a fancy name. Basically, differentiate each term and solve for y'. Anything with a y will need a y'. To do this, expand everything to remove the denominator:
x^2=(x+y)(y^2+8)=xy^2+8x+y^3+8y.
Note we have a term with x and y. For this we'll need to use the product rule where D(uv)=uD(v)+vD(u)
Then D(xy^2)=x(2yy')+y^2
D(x^2)=2x
D(8x)=8
D(y^3)=2y^2y'
D(8y)=8y'
2x=2xyy'+y^2+8+2y^2y'+8y'
Isolate all terms with y':
2x-8-y^2=y'(2xy+2y^2+8)
y'=(2x-8-y^2)/(2xy+2y^2+8)
Hope that helps!
