1. Problem: Find $\frac{dy}{dx}$ if $x + y^4 = 10$ and $y \neq 0$. 2. Differentiate both sides with respect to $x$: $$\frac{d}{dx}(x) + \frac{d}{dx}(y^4) = \frac{d}{dx}(10)$$ This gives: $$1 + 4y^3 \frac{dy}{dx} = 0$$ 3. Solve for $\frac{dy}{dx}$: $$4y^3 \frac{dy}{dx} = -1$$ $$\frac{dy}{dx} = \frac{-1}{4y^3}$$ 4. Therefore, the correct option is (A) $\frac{-1}{4y^3}$. --- 5. Problem: If $h(x) = \int_0^{x^2} g(t) dt$, find $h'(x)$. 6. Using the Fundamental Theorem of Calculus and chain rule: $$h'(x) = g(x^2) \cdot \frac{d}{dx}(x^2) = g(x^2) \cdot 2x = 2x g(x^2)$$ 7. The correct option is (C) $2x g(x^2)$. --- 8. Problem: Given $f(f(x)) = x$, which graph could represent $f$? 9. Explanation: The condition $f(f(x)) = x$ means $f$ is an involution and its own inverse. The graph of $f$ must be symmetric about the line $y = x$. 10. Option (D) describes a function resembling a hyperbola with branches in the first and third quadrants, which is symmetric about $y=x$. 11. Therefore, option (D) is the correct graph for $f$. Final answers: 1) $\frac{dy}{dx} = \frac{-1}{4y^3}$ 2) $h'(x) = 2x g(x^2)$ 3) Graph: option (D)