Implicit Differentiation Process

The derivative of $x$ with respect to x is 1 and that of $y$ is $dy/dx$.

Collecting like terms involving $dy/dx$ allows solving for the derivative.

The derivative of $\sin(x*y)$ is simply $\cos(x*y)$.

Differentiating $\sin(x*y)$ requires the chain rule, leading to $\cos(x*y)*(y + x*(dy/dx))$.