Question

PLEASE ANSWER ALL PARTS AND SHOW ALL OF YOUR WORK NEATLY.

Suppose ( f(x, y)=frac{x}{y}, P=(-4,2) ) and ( mathbf{v}=2 mathbf{i}+3 mathbf{j} ). A. Find the gradient of ( f ). [ (nabla f)(x, y)=frac{1}{y} mathbf{i}+frac{-x}{y²mathbf{j} ] N

Answer 1

The gradient of the function f(x, y) is (1/y)i - (x/y^2)j.

Step-by-step explanation:

To find the gradient of the function f(x, y) = x/y, we need to find the partial derivatives with respect to x and y. Taking the partial derivative with respect to x gives us (∂f/∂x) = 1/y. Taking the partial derivative with respect to y gives us (∂f/∂y) = -x/y^2. Therefore, the gradient of f(x, y) is given by ∇f(x, y) = (1/y)i - (x/y^2)j.