נגזרות של פונקציות וקטוריות

Imagine a car rounding a mountain curve. At time t, its position is at point P. After a brief interval h, it reaches point Q at time t + h.

The vector from P to Q shows the car's overall change in position during that time and is called a secant vector. It connects two distinct positions along the car's route.

As the time interval h gets smaller, point Q moves closer to point P.

The secant vector starts to reflect the road's direction at P.

Now multiply this secant vector by one over h.

This operation doesn’t change the direction; it only scales the vector’s length.

As h tends to zero, this scaled vector approaches a vector with a well-defined direction and magnitude.

That limiting vector is called the derivative vector, or the tangent vector r′(t).

The vector r′(t) is tangent to the curve at the point P. It represents the instantaneous velocity, both the speed and the direction of the car at any time t.