Consider the following question. Question

I know that the gradient is perpendicular to the tangent plane (and hence parallel to the normal vector). But how do I find the gradient vector and the equation of the tangent plane at the point (-1, 1, 0) from this functional equation? The Chain Rule doesn't seem to help.


Another important property of gradients you need to consider to solve this problem is that they are perpendicular to the level surfaces. You have a level surface where $f=1$, which I'll name $P$.

So in this case, you just need to find the normal of $P$ when $x=-1$ and $z = 0$, which can be done easily by taking its partial derivatives and finding their cross product: $$P=\langle x, 2x+3z+3, z\rangle\\\frac{\partial P}{\partial x} = \langle1, 2, 0\rangle\\\frac{\partial P}{\partial z}=\langle0, 3, 1\rangle\\P_x\times P_z = \langle2, -1, 3\rangle.$$

So it actually turns out that in this problem, the gradient is independent of $x$ and $z$.


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