Use implicit differentation to determine a formula for the derivative #dy/dx#:
#d/dx (sin^2 y) - d/dx (cos^2 x^2) = d/dx C#
#2 (sin y) * (cos y) dy/dx - 2 (cos x^2) * (-sin x^2) * (2x) = 0#
# (sin 2y) dy/dx + 4x * (cos x^2) * (sin x^2) = 0 #
# (sin 2y) dy/dx + 2x * (sin (2x^2)) = 0 #
# (sin 2y) dy/dx = -2x (sin (2x^2)) #
#dy/dx = (-2x (sin (2x^2))) / (sin 2y)#
To get the slope, we then evaluate this at the point #(pi/3, pi/3)#:
#(dy/dx)|_(pi/3, pi/3) = (-2 * (pi/3) * (sin (2*(pi/3)^2))) / (sin (2*pi/3))#
# = ((- (2pi)/3) * (sin ((2pi^2)/9)))/(sin((2pi)/3))=((- (2pi)/3) * (sin ((2pi^2)/9)))/(sqrt(3)/2)#
# = ((- (4pi)/3) * (sin ((2pi^2)/9)))/(sqrt(3))= ((- (4pi sqrt(3))/3) * (sin ((2pi^2)/9)))/3#
# = (- (4pi sqrt(3))/9) * (sin ((2pi^2)/9))#
#~~-1.964#
