I stumbled upon this calculus implicit differential question: find $\frac{{\textstyle du}}{{\textstyle dy}}$ of the function $u=\mathrm{sin}({y}^{2}+u)$.

The answer is $\frac{{\textstyle 2y\mathrm{cos}({y}^{2}+u)}}{{\textstyle 1-\mathrm{cos}({y}^{2}+u)}}$. I understand how to get the answer for the numerator, but how do we get the denominator part?

And can anyone point out the real intuitive difference between chain rule and implicit differentiation? I can't seem to get my head around them and when or where should I use them.

The answer is $\frac{{\textstyle 2y\mathrm{cos}({y}^{2}+u)}}{{\textstyle 1-\mathrm{cos}({y}^{2}+u)}}$. I understand how to get the answer for the numerator, but how do we get the denominator part?

And can anyone point out the real intuitive difference between chain rule and implicit differentiation? I can't seem to get my head around them and when or where should I use them.