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.
