1. **State the problem:** Given the function $f(x,y) = 6x^2 \sin(x + y^2)$, find the mixed partial derivative $f_{xy}$ and evaluate it at $x=1$, $y=7$. 2. **Recall the formula and rules:** - The mixed partial derivative $f_{xy}$ means first differentiate $f$ with respect to $x$, then differentiate the result with respect to $y$. - Use the product rule and chain rule for differentiation. 3. **Find $f_x$: differentiate $f$ with respect to $x$:** $$f_x = \frac{\partial}{\partial x} \left(6x^2 \sin(x + y^2)\right) = 6 \cdot \frac{\partial}{\partial x} \left(x^2 \sin(x + y^2)\right)$$ Using product rule: $$f_x = 6 \left(2x \sin(x + y^2) + x^2 \cos(x + y^2) \cdot \frac{\partial}{\partial x}(x + y^2)\right)$$ Since $\frac{\partial}{\partial x}(x + y^2) = 1$: $$f_x = 6 \left(2x \sin(x + y^2) + x^2 \cos(x + y^2)\right)$$ 4. **Find $f_{xy}$: differentiate $f_x$ with respect to $y$:** $$f_{xy} = 6 \frac{\partial}{\partial y} \left(2x \sin(x + y^2) + x^2 \cos(x + y^2)\right)$$ Since $x$ is constant with respect to $y$: $$f_{xy} = 6 \left(2x \cos(x + y^2) \cdot \frac{\partial}{\partial y}(x + y^2) + x^2 \cdot \frac{\partial}{\partial y} \cos(x + y^2)\right)$$ Calculate derivatives: $$\frac{\partial}{\partial y}(x + y^2) = 2y$$ $$\frac{\partial}{\partial y} \cos(x + y^2) = -\sin(x + y^2) \cdot 2y$$ Substitute: $$f_{xy} = 6 \left(2x \cos(x + y^2) \cdot 2y + x^2 (-\sin(x + y^2) \cdot 2y)\right)$$ Simplify: $$f_{xy} = 6 \left(4xy \cos(x + y^2) - 2x^2 y \sin(x + y^2)\right)$$ 5. **Evaluate $f_{xy}$ at $x=1$, $y=7$:** Calculate inside the trig functions: $$x + y^2 = 1 + 7^2 = 1 + 49 = 50$$ Calculate: $$f_{xy}(1,7) = 6 \left(4 \cdot 1 \cdot 7 \cos(50) - 2 \cdot 1^2 \cdot 7 \sin(50)\right) = 6 \left(28 \cos(50) - 14 \sin(50)\right)$$ 6. **Approximate values:** Using a calculator or approximation tool: $$\cos(50) \approx 0.96497$$ $$\sin(50) \approx -0.26237$$ Substitute: $$28 \times 0.96497 = 27.0192$$ $$14 \times (-0.26237) = -3.6732$$ So: $$f_{xy}(1,7) = 6 (27.0192 - (-3.6732)) = 6 (27.0192 + 3.6732) = 6 \times 30.6924 = 184.1544$$ **Final answer:** $$f_{xy}(1,7) \approx 184.15$$