1. Stating the problem: Find the derivative of the function $$y = x^2 \sin^4 x + x \cos^{-2} x.$$\n\n2. Rewrite the function for clarity: $$y = x^2 (\sin x)^4 + x (\cos x)^{-2}.$$\n\n3. Use the sum rule and differentiate term-by-term:\nLet $$u = x^2 (\sin x)^4$$ and $$v = x (\cos x)^{-2}.$$\nSo, $$\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}.$$\n\n4. Differentiate $$u = x^2 (\sin x)^4$$ using the product rule:\n$$\frac{du}{dx} = 2x (\sin x)^4 + x^2 \cdot 4 (\sin x)^3 \cos x$$\nHere, \(\frac{d}{dx}(\sin x)^4 = 4 (\sin x)^3 \cos x\) by chain rule.\n\n5. Differentiate $$v = x (\cos x)^{-2}$$ using product rule:\n$$\frac{dv}{dx} = 1 \cdot (\cos x)^{-2} + x \cdot (-2)(\cos x)^{-3} (-\sin x)$$\nSimplify:\n$$\frac{dv}{dx} = (\cos x)^{-2} + 2x (\cos x)^{-3} \sin x.$$\n\n6. Combine to get the full derivative:\n$$\frac{dy}{dx} = 2x (\sin x)^4 + 4x^2 (\sin x)^3 \cos x + (\cos x)^{-2} + 2x (\cos x)^{-3} \sin x.$$\n\n7. This is the derivative of the given function.