Find the gradient of the function f(x,y)=x^3 + y^3 -3xy at the point (2,1), given that f(2,1) = 6.

Firstly, establish that the correct method to do this is via differentiation: specifically implicit differentiation. To find the gradient, we need to find dy/dx. The differential with respect to x of x³ = 3x². The differential with respect to x of y³ = 3y²dy/dx. The differential with respect to x of -3xy = -3y - 3xdy/dx (By Chain Rule - u = -3x v = y.) The differential with respect to x of 6 = 0. As such, we can form the equation: 0 = 3x² + 3y²dy/dx - 3y - 3xdy/dx. Which can be rearranged to give dy/dx = (3x² - 3y)/(3y² - 3x). Subbing in our values for x and y, we get dy/dx = (32² - 31)/(31² - 32) = (12 - 3)/(3 - 6) = 9/-3 = -3. Thus our solution is -3.