The "directional derivative" of a function is simply the rate of change of a function in the given direction. The gradient of a function is the directional derivative in the direction of [g]greatest[/b] increase. There is a directional derivative in every direction and the gradient is one of them. Of course, the "directional derivative" opposite to grad f is the fastest decrease and is just the negative of grad f. Exactly perpendicular to them the directional derivative is 0.i thought the direction derivative is tangent to a surface concerns and hence perpendicular to grad at a point on the surface?
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What your book is doing, given the surface, z= f(x,y), is think of that as a "level surface" of the function F(x,y,z)= z- f(x,y). The surface itself satisfies F(x, y, z)= 0, a constant, so a directional derivative in a direction tangent to the surface is 0 and is perpendicular to the surface.