What's the PROOF of the Directional Derivative of a 3-variable function?
DuF(x,y,z)=(Fx,Fy,Fz).(U1,U2,U3) where U=(U1,U2,U3) is a unit vector.
Assuming that F is differentiable at the given point, then it can be approximated by the linear function . (That follows from the definition of "differentiable at a point")
The line through , in the direction of , is given by the parametric equations , , .
To find the derivative of F in the direction U, replace x, y, and z with those in the approximation for F, to get , a linear function of t with derivative equal to its slope, .