Question

If f(x,y,z)=(x2y2−2y3z2−4x3z2+xyz) , the gradient of f at (1,−2,−1) is

Answer 1

$\frac{\partial f}{\partial x}=2xy^2-12x^2z^2+yz.$

$\frac{\partial f}{\partial y}=2x^2y-6y^2z^2+xz.$

$\frac{\partial f}{\partial z}=-4y^3z-8x^3z+xy.$

At the point (1,-2,-1): $\frac{\partial f}{\partial x}=-6, \;\frac{\partial f}{\partial y}=-29,\; \frac{\partial f}{\partial z}=-26.$

Gradient: $\nabla =[-6,-29,-26].$