find the gradient for the follwing function at (1,2,-1):-f(x,y,z)=xy2/z3

Question

function question

Answer 1

The gradient is a 3-dimensional vector.

If the 4-dimensional function f(x,y,z)=xy²/z³ the gradient components are:

∂f/∂x=y²/z³, at (1,2,-1) this is -4;

∂f/∂y=2xy/z³, at (1,2,-1) this is -4;

∂f/∂z=-3xy²/z⁴, at (1,2,-1) this is -12.

The gradient is <-4,-4,-12> which can be written -4i-4j-12k where i, j, k are Cartesian unit vectors.