QuestionFind the maximum rate of change of $f$ at the given point and the direction in which it occurs. $$f(x, y)=\sin (x y), \quad(9,0)$$
Step 1 Recall that the direction in which the maximum rate of change of $f(x, y)$ occurs at a point $(a, b)$ is given by the vector $\nabla f(a, b)$. For $f(x, y)=\sin (x y)$, we have $$\nabla f(x, y)=\square$$

Studdy Solution
The maximum rate of change is the magnitude of the gradient vector at $(9, 0)$:
$$\|\nabla f(9, 0)\| = \sqrt{0^2 + 9^2} = 9$$
The direction of the maximum rate of change is given by the unit vector in the direction of $\nabla f(9, 0)$:
Since $\nabla f(9, 0) = (0, 9)$, the direction is simply $(0, 1)$.
The maximum rate of change of $f$ at the point $(9, 0)$ is $\boxed{9}$, and it occurs in the direction of the vector $\boxed{(0, 1)}$.