For this problem, we are given an equation and we need to calculate its implicit differentiation.
The equation is:
[tex]\sin(xy)=y^2[/tex]
We have:
[tex]\begin{gathered} \sin(x\cdot y)=y(x)^2\\ \\ \frac{d(\sin(x\cdot y))}{dx}=\frac{d(y^2)}{dx}\\ \\ (x\cdot y)^{\prime}\cdot\cos(x\cdot y)=2\cdot y\cdot y^{\prime}\\ \\ y\cdot cos(xy)+xy^{\prime}\cdot cos(xy)=2yy^{\prime}\\ \\ 2yy^{\prime}-xy^{\prime}\cdot\cos(xy)=y\cdot\cos(xy)\\ \\ y^{\prime}(2y-x\cdot\cos(xy))=y\cdot\cos(xy)\\ \\ y^{\prime}=\frac{y\cdot\cos(xy)}{2y-x\cdot\cos(xy)} \end{gathered}[/tex]
