Implicit Differentiation Procedure

Differentiate both sides with respect to $x$, Apply the chain rule to $\cos(x*y)$ yielding $-\sin(x*y)*(x*(dy/dx) + y)$ using the product rule for $x*y$, Differentiate the right side to obtain $1 + dy/dx$, Collect all terms containing $dy/dx$ and solve for it

Differentiate both sides, but mistakenly treat $x*y$ as separable without the product rule, then combine terms improperly, resulting in an incorrect ordering

Differentiate the right side first as $1$, then differentiate $\cos(x*y)$ ignoring the negative sign, and finally set the derivatives equal

Apply the product rule on the right side instead of the left, differentiate $\cos(x*y)$ as if it were a constant, and then solve for $dy/dx$