My book states without proof that the directional derivative at any point is orthogonal to the tangent to the level set at the same point.
I don't even know where to get started.
All I contribute is that :
Assume $f : R^n \rightarrow R$ (I can make this assumption as per question)
$D_v f(a_1,a_2,\cdots,a_n) = ||\nabla{f(x,y)}||_{a_1,a_2,\cdots,a_n} \cdot \dfrac{v}{||v||}$
I need to show that 2 vectors are orthogonal and thus, I feel that there is a point where I'd need to show that the inner product of the above directional derivative vector with the tangent vector is 0. (But what is the tangent vector?)