Limit and derivatives

Dear Student,

\cos^{3} x + \sin^{2} (x y) = 5 a^{2} + y^{3} 3 \cos^{2} x . \frac{d}{d x} (\cos x) + 2 \sin (x y) . \frac{d}{d x} (\sin x y) = 0 + 3 y^{2} \frac{d y}{d x} - 3 \sin x . \cos^{2} x + 2 \sin (x y) \cos (x y) . \frac{d}{d x} (x y) = 3 y^{2} \frac{d y}{d x} - 3 \sin x . \cos^{2} x + 2 \sin (x y) \cos (x y) [x \frac{d y}{d x} + y] = 3 y^{2} \frac{d y}{d x} S i n c e, 2 \sin A \cos A = \sin 2 A - 3 \sin x . \cos^{2} x + \sin (2 x y) [x \frac{d y}{d x} + y] = 3 y^{2} \frac{d y}{d x} - 3 \sin x . \cos^{2} x + x \sin (2 x y) \frac{d y}{d x} + y \sin (2 x y) = 3 y^{2} \frac{d y}{d x} - 3 \sin x . \cos^{2} x + y \sin (2 x y) = 3 y^{2} \frac{d y}{d x} - x \sin (2 x y) \frac{d y}{d x} y \sin (2 x y) - 3 \sin x . \cos^{2} x = \frac{d y}{d x} [3 y^{2} - x \sin (2 x y)] \frac{d y}{d x} = \frac{y \sin (2 x y) - 3 \sin x . \cos^{2} x}{3 y^{2} - x \sin (2 x y}

Regards!