Find the equation of the line for this impilcit differential

Question

Here is the question.

A set of points in this graph that satisfies the the equation of the line tangent to this curve at the point (0,4)

So I started by finding the derivative

But I am not sure what to do next.

Answer 1

This is too long for a comment but I suppose it could be of some interest.

You could have obtained $y'$ much faster considering the function $$F=x \cos(x y)+y-4=0$$ and used the implicit function theorem $$F'_x=\cos (x y)-x y \sin (x y)$$ $$F'_y=1-x^2 \sin (x y)$$ $$y'=\frac{dy}{dx}=-\frac{F'_x}{F'_y}=\frac{x y \sin (x y)-\cos (x y)}{1-x^2 \sin (x y)}$$ Now, as Mirko commented, just plug $x=0$ to see that, for any $y$, $y'=-1$.