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.
DD
Related Maths A Level answers
All answers ▸The equation x^3 - 3*x + 1 = 0 has three real roots; Show that one of the roots lies between −2 and −1
Circle C has equation x^2 + y^2 - 6x + 4y = 12, what is the radius and centre of the circle
You are given the equation of the line y=x^3+x^2-2x. Find the stationary points of the curve and determine the maximum and minimum points and find where it crosses the x-axis and thus sketch the graph
Use the substitution u=2+ln(t) to find the exact value of the antiderivative of 1/(t(2+ln(t))^2)dt between e and 1.