1. We are asked to find the derivative of the function $$f(x) = \sin^2 x + \frac{4}{\sin^2 x}$$ with respect to $$x$$. 2. Recall the derivative rules: - The derivative of $$\sin x$$ is $$\cos x$$. - Use the chain rule for $$\sin^2 x = (\sin x)^2$$: $$\frac{d}{dx} (u^2) = 2u \frac{du}{dx}$$. - For the term $$\frac{4}{\sin^2 x}$$, rewrite as $$4 \sin^{-2} x$$ and use the power rule combined with the chain rule. 3. Differentiate each term: - $$\frac{d}{dx} \sin^2 x = 2 \sin x \cos x$$. - $$\frac{d}{dx} 4 \sin^{-2} x = 4 \cdot (-2) \sin^{-3} x \cos x = -8 \frac{\cos x}{\sin^3 x}$$. 4. Combine the derivatives: $$f'(x) = 2 \sin x \cos x - 8 \frac{\cos x}{\sin^3 x}$$. 5. Factor out $$2 \cos x$$: $$f'(x) = 2 \cos x \left( \sin x - \frac{4}{\sin^3 x} \right)$$. This is the derivative of the given function. Final answer: $$\boxed{f'(x) = 2 \cos x \left( \sin x - \frac{4}{\sin^3 x} \right)}$$