Finding a unit normal vector to a surface

Question

Find a unit normal vector to the surface z(x,y,z)=0 at the point P(-4,-5,9) for the function

f(x,y,z)=ln(-4x/(-5y-z))

Please write your answer as a vector ⟨a,b,c⟩ with a negative z component, and show your answer accurate to 4 decimal places

Answer 1

The gradient of f(x,y,z) is normal to any surface f(x,y,z) = constant.

So, calculate the gradient of f(x,y,z) at point P and normalize it to get a unit vector out of it.

If necessary, multiply the unit vector by -1, to get a negative z component, as the problem is asking.

Answer 2