1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ given $y = u^5 - 2u^3 + 8$ and $u = x^2 + 1$. 2. **Formula and rules:** Use the chain rule for derivatives: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ 3. **Find $\frac{dy}{du}$:** Differentiate $y$ with respect to $u$: $$\frac{dy}{du} = 5u^4 - 6u^2$$ 4. **Find $\frac{du}{dx}$:** Differentiate $u$ with respect to $x$: $$\frac{du}{dx} = 2x$$ 5. **Apply the chain rule:** $$\frac{dy}{dx} = (5u^4 - 6u^2) \cdot 2x$$ 6. **Substitute back $u = x^2 + 1$:** $$\frac{dy}{dx} = (5(x^2 + 1)^4 - 6(x^2 + 1)^2) \cdot 2x$$ 7. **Final answer:** $$\boxed{\frac{dy}{dx} = 2x \left(5(x^2 + 1)^4 - 6(x^2 + 1)^2\right)}$$