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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.

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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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