doesn't the normal plane defines all the tangents or directional derivatives at the point t of a 3 dimensional curve?
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I'm not sure what you mean about directional derivatives; a curve is a function of just one variable, so we can't really take directional derivatives, and there is only one tangent, which is the derivative.
Think about it: a normal plane has to be perpendicular to the curve. What vector could we use as the normal vector for the plane? It should be tangent to the curve, because if the vector is tangent to the curve and the plane is normal to the vector, then the plane will be normal to the curve. Well, the derivative works as a tangent vector.