Any tangent line to the curve has a slope given by [tex]\dfrac{\mathrm dy}{\mathrm dx}[/tex].
[tex]\cos y=x-4y\implies-\sin y\,\dfrac{\mathrm dy}{\mathrm dx}=1-4\,\dfrac{\mathrm dy}{\mathrm dx}[/tex]
[tex]\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac1{4-\sin y}[/tex]
Since [tex]|\sin y|\le1[/tex], the denominator ranges from 3 to 5, so the range of slopes is [tex]\dfrac15\le\dfrac{\mathrm dy}{\mathrm dx}\le\dfrac13[/tex].
