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 \( \nabla g(x, y)=\langle\square, \square \)

To compute the gradient of the function \( g(x, y) = x^{2} - 8x^{2}y - 6xy^{2} \), we need to find the partial derivatives with respect to \( x \) and \( y \). The partial derivative with respect to \( x \) is: \[ \frac{\partial g}{\partial x} = 2x - 16xy - 6y^{2} \] The partial derivative with respect to \( y \) is: \[ \frac{\partial g}{\partial y} = -8x^{2} - 12xy \] Thus, the gradient can be expressed as: \[ \nabla g(x, y) = \left\langle 2x - 16xy - 6y^{2}, -8x^{2} - 12xy \right\rangle \] Now evaluating the gradient at point \( P(-1, 3) \): For \( x = -1 \) and \( y = 3 \): \[ \frac{\partial g}{\partial x} \bigg|_{P} = 2(-1) - 16(-1)(3) - 6(3)^{2} = -2 + 48 - 54 = -8 \] \[ \frac{\partial g}{\partial y} \bigg|_{P} = -8(-1)^{2} - 12(-1)(3) = -8 + 36 = 28 \] So, the gradient at point \( P \) is: \[ \nabla g(-1, 3) = \langle -8, 28 \rangle \] To summarize, the gradient is \( \nabla g(x, y) = \langle -8, 28 \rangle \).