Compute the gradient of the following function and evaluate it at the given point \( P \), \( g(x, y)=x^{2}-8 x^{2} y-6 x y^{2} ; P(-1,3) \) The gradient is \( \nabla g(x, y)=\langle\square, \square \).

We start with the function \[ g(x, y)=x^{2}-8x^{2}y-6xy^{2}. \] **Step 1. Compute the partial derivative with respect to \( x \):** Differentiate each term treating \( y \) as a constant. 1. \(\frac{\partial}{\partial x}(x^2) = 2x\). 2. \(\frac{\partial}{\partial x}(-8x^2y) = -8y \cdot 2x = -16xy\). 3. \(\frac{\partial}{\partial x}(-6xy^2) = -6y^2\). Thus, \[ \frac{\partial g}{\partial x} = 2x - 16xy - 6y^2. \] **Step 2. Compute the partial derivative with respect to \( y \):** Differentiate each term treating \( x \) as a constant. 1. \(\frac{\partial}{\partial y}(x^2) = 0\). 2. \(\frac{\partial}{\partial y}(-8x^2y) = -8x^2\). 3. \(\frac{\partial}{\partial y}(-6xy^2) = -6x \cdot 2y = -12xy\). Thus, \[ \frac{\partial g}{\partial y} = -8x^2 - 12xy. \] **Step 3. Write the gradient vector:** The gradient is given by \[ \nabla g(x, y)=\langle 2x - 16xy - 6y^2,\; -8x^2 - 12xy \rangle. \] **Step 4. Evaluate the gradient at \( P(-1,3) \):** Substitute \( x = -1 \) and \( y = 3 \) into each component. *For the \( x \)-component:* \[ 2(-1) - 16(-1)(3) - 6(3)^2 = -2 + 48 - 54 = -8. \] *For the \( y \)-component:* \[ -8(-1)^2 - 12(-1)(3) = -8 + 36 = 28. \] Thus, at the point \( P(-1,3) \), \[ \nabla g(-1,3)=\langle -8,\; 28 \rangle. \]