Taking the derivative of both sides with respect to [tex]x[/tex] gives
[tex]\dfrac{\mathrm d}{\mathrm dx}[x^2]=\dfrac{\mathrm d}{\mathrm dx}[-2+y+5\cos y][/tex]
[tex]\implies2x=\dfrac{\mathrm dy}{\mathrm dx}-5\sin y\,\dfrac{\mathrm dy}{\mathrm dx}[/tex]
[tex]\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2x}{1-5\sin y}[/tex]
When [tex]y=11[/tex], we have two possible values of [tex]x[/tex]:
[tex]x^2=-2+11+5\cos11\implies x=\pm\sqrt{9+5\cos11}\approx\pm3.004[/tex]
so we have two possible values [tex]\dfrac{\mathrm dy}{\mathrm dx}\approx\pm1.001[/tex].
