Calculus: Early Transcendentals, Chapter 3, 3.5, Section 3.5, Problem 12

Note:- 1) If y = cosx ; then dy/dx = -sinx
2) If y = sinx ; then dy/dx = cosx
3) If y = u*v ; where both u & v are functions of 'x' , then
dy/dx = u*(dv/dx) + v*(du/dx)
4) If y = k ; where 'k' = constant ; then dy/dx = 0
Now, the given function is :-
cos(x*y) = 1 + siny
Differentiating both sides w.r.t 'x' we get
-sin(xy)*[y + x*(dy/dx)] = cosy*(dy/dx)
or, -y*sin(xy) -x*sin(xy)*(dy/dx) = cosy*(dy/dx)
or, -y*sin(xy) = (dy/dx)*[cosy + x*sin(xy)]
or, dy/dx = [-y*sin(xy)]/[cosy + x*sin(xy)]