Final answer:
The gradient of the function f(x,y) = x/y at the point p(4, -3) is the vector ∇f(4, -3) = (-1/3, -4/9).
Step-by-step explanation:
The student has asked to find the gradient of the function f(x,y) = x/y at the point p(4,-3). To find the gradient of a two-variable function, we calculate the partial derivatives with respect to both variables. The gradient is a vector that points in the direction of the steepest ascent.
For the given function f(x,y) = x/y, the partial derivative with respect to x, denoted as f_x, is 1/y, and the partial derivative with respect to y, denoted as f_y, is -x/y². At the point (4, -3), these partial derivatives are f_x(4, -3) = -1/3 and f_y(4, -3) = -4/9.