Question

E^y*cos x = 1 + sin(xy)

Answer 1

I assume you are studying implicit differentiation. Solving for dy/dx.
For the Left side use product rule. For Right side, use chain rule. Remember any time you take derivative of "y-term" you must multiply it by dy/dx.

--->
`(dy)/(dx) e^y cos (x) - e^y sin(x) = (y + x (dy)/(dx)) cos (xy)`

Get dy/dx terms on one side:

`(dy)/(dx) e^y cos (x) - (dy)/(dx) x cos(xy) = e^y sin(x) + y cos (xy)`

Factor out dy/dx and solve:

`(dy)/(dx) (e^y cos (x) - x cos(xy)) = e^y sin(x) + y cos (xy) \\.......\\ (dy)/(dx) = (e^y sin(x) + y cos (xy) )/(e^y cos (x) - x cos(xy))`