Question

in what direction is the derivative of ( ) 2 f x, y = xy − y at the point (3,2) equal to zero? give the answer as a unit vector.

Answer 1

To find the direction of the derivative at the point (3, 2) where it is equal to zero, we need to find the gradient of the function at that point and then find a unit vector in that direction.

Taking partial derivatives with respect to x and y, we get:

∂f/∂x = y - 0 = y

∂f/∂y = x - 1

At the point (3, 2), we have:

∂f/∂x = 2

∂f/∂y = 3

So the gradient at that point is the vector <2, 3>.

To find a unit vector in the direction of this gradient, we first calculate its magnitude:

|<2, 3>| = sqrt(2^2 + 3^2) = sqrt(13)

Then we divide the gradient vector by its magnitude:

<2, 3>/sqrt(13) = (2/sqrt(13))i + (3/sqrt(13))j

So the unit vector in the direction of the derivative at the point (3, 2) is (2/sqrt(13))i + (3/sqrt(13))j.

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