Assuming that Fisdifferentiable 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, .