1. The problem states that for a differentiable function $y(x)$, the equation $x + y^4 = 10$ holds with $y \neq 0$. We want to find $\frac{dy}{dx}$. 2. Differentiate both sides implicitly with respect to $x$. The derivative of $x$ is 1, and using the chain rule, the derivative of $y^4$ is $4y^3 \frac{dy}{dx}$. So we get: $$1 + 4y^3 \frac{dy}{dx} = 0$$ 3. Solve for $\frac{dy}{dx}$: $$4y^3 \frac{dy}{dx} = -1$$ $$\frac{dy}{dx} = \frac{-1}{4y^3}$$ 4. Hence, the correct answer is (C) $\frac{dy}{dx} = -\frac{x}{4y^3}$? Wait, options do not perfectly match; check again. Actually, from the original equation $x + y^4 = 10$, so $x = 10 - y^4$. When differentiating, treat $y$ as a function of $x$. Implicit differentiation: $$\frac{d}{dx}(x) + \frac{d}{dx}(y^4) = \frac{d}{dx}(10)$$ $$1 + 4y^3 \frac{dy}{dx} = 0$$ $$4y^3 \frac{dy}{dx} = -1$$ $$\frac{dy}{dx} = -\frac{1}{4y^3}$$ Since $x + y^4 = 10$ is given, the derivative does not depend on $x$, so answer (C) $-\frac{x}{4y^3}$ is incorrect. The exact derivative is $-\frac{1}{4y^3}$. Thus answer (A) is $\frac{1}{4y^3}$ (positive), so no. Answer (B) is $\frac{1}{y^4}$ no. Answer (C) $-\frac{x}{4y^3}$ no. Answer (D) $\frac{9}{4y^3}$ no. Answer (E) $\frac{10 - x}{y^4}$ no. The closest is (none) but derivative is $-\frac{1}{4y^3}$. We'll choose derivative as $-\frac{1}{4y^3}$ and note that the correct choice is not given but the calculation is as above. --- 5. We have $h(x) = \int_0^{x^2} g(t) dt$ and want $h'(x)$. Apply the Fundamental Theorem of Calculus and chain rule: $$h'(x) = g(x^2) \cdot \frac{d}{dx}(x^2) = g(x^2) \cdot 2x = 2x g(x^2)$$ Hence the correct answer is (C). --- 6. Given $f(f(x)) = x$ for all $x$ in the domain, $f$ is an involution, and therefore its graph is symmetric about the line $y=x$. Graphically, such a function is typically strictly increasing and passes through the origin with symmetry about $y=x$. Check options: (A) strictly increasing curved function with origin pass and range from near 0 to 2 for x from negative to positive. This could be an involution. (B) strictly decreasing - fails symmetry for involution (C) cubic-like with local min at origin - not typical for involution (D) curve with asymptotes - unlikely (E) strictly decreasing past origin - no Hence (A) is the plausible graph for $f$. --- Final answers: - Q4 derivative: $\boxed{\frac{dy}{dx} = -\frac{1}{4y^3}}$ (no matching option) - Q5 derivative: $\boxed{h'(x) = 2x g(x^2)}$ - Q6 graph: (A)