1. The problem asks to find a unit vector in the direction where the function $f(x,y) = 6x^6 y^7$ increases most rapidly at the point $P(-1,1)$, and to find the rate of change of $f$ at $P$ in that direction. 2. The direction of the greatest increase of a function is given by the gradient vector $\nabla f(x,y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$. 3. Calculate the partial derivatives: $$\frac{\partial f}{\partial x} = 6 \cdot 6 x^{5} y^{7} = 36 x^{5} y^{7}$$ $$\frac{\partial f}{\partial y} = 6 x^{6} \cdot 7 y^{6} = 42 x^{6} y^{6}$$ 4. Evaluate the gradient at $P(-1,1)$: $$\nabla f(-1,1) = (36 (-1)^{5} (1)^{7}, 42 (-1)^{6} (1)^{6}) = (36 \cdot (-1) \cdot 1, 42 \cdot 1 \cdot 1) = (-36, 42)$$ 5. Find the magnitude of the gradient vector: $$||\nabla f(-1,1)|| = \sqrt{(-36)^2 + 42^2} = \sqrt{1296 + 1764} = \sqrt{3060} = 6 \sqrt{85}$$ 6. The unit vector in the direction of greatest increase is: $$\mathbf{u} = \frac{1}{||\nabla f(-1,1)||} \nabla f(-1,1) = \frac{1}{6 \sqrt{85}} (-36, 42) = \left(-\frac{6}{\sqrt{85}}, \frac{7}{\sqrt{85}}\right)$$ 7. The rate of change of $f$ at $P$ in that direction is the magnitude of the gradient: $$6 \sqrt{85}$$