[Math] Find max rate of change of $f(x,y) = \sin(xy)$ at the point $(1,0)$ and in the direction in which it occurs.

Question

Find max rate of change of $f(x,y) = \sin(xy)$ at the point $(1,0)$ and in the direction in which it occurs.

I did the following:

$$\nabla f = <\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}>$$

$$= <y\cos(xy), x\cos(xy)>$$

$$\nabla f_{(1,0)} = <(0)\cos(0),\cos(0)>$$

$$= <0,1>$$

This vector tells me both the direction and the magnitude in which the maximum rate of change occurs because it happens to already be a unit vector, no?

But shouldn't I somehow get a scalar for the maximum rate of change?

Answer 1

Hint:

The magnitude of the gradient is a scalar.