For the surface given below, find the tangent plane at point P and the normal line at P

Question

$$xy+2yz-xz^2+10=0$$ $$P(-5,5,1) $$ The problem is i couldn't figure out that am I suppose to find partial derivatives with implicit differentiation for gradient vector or not. Any help will be mean so much to me.Thank you.

Answer 1

Note that $F(x,y,z)=0$ define a level surface then we can calculate the gradient, which is orthogonal to the level surface, and take $\vec n = (a,b,c)=\nabla F(x,y,z)$ and thus determine also the tangent plane $ax+by+cz+d=0$ (with d to be found by the condition that P belongs to the tangent plane).