Derivatives of Vector Functions

A secant vector connects two distinct positions along a curve, representing the overall change in position during a time interval. For a car at position P at time t and position Q at time t + h, the secant vector is r(t+h) − r(t). This vector provides a crude estimate of the direction of motion and captures displacement over the interval.

Average velocity is calculated by scaling the secant vector by the reciprocal of the time interval h, forming the difference quotient [r(t+h) − r(t)]/h. This operation preserves the secant vector's direction while adjusting its magnitude to reflect average speed over the interval, providing a measure of how fast the object moved during that time period.

As the time interval h approaches zero, point Q converges to point P, and the secant vector becomes infinitesimally short. The scaled secant vector approaches a limiting vector with a well-defined direction and magnitude, which is the derivative vector or tangent vector r'(t).

The derivative vector r'(t) is tangent to the curve at a point and represents instantaneous velocity, including both speed and direction of motion at any time t. For a car on a curved road, it shows exactly how fast and in what direction the car is moving at that precise moment, making it essential for understanding motion in space velocity and acceleration.

The derivative vector is evaluated component-wise by taking the limit of each component's difference quotient separately. For r(t) = ⟨x(t), y(t), z(t)⟩, the derivative is r'(t) = ⟨x'(t), y'(t), z'(t)⟩, where each component is the ordinary derivative of its corresponding position function.

The derivative vector is tangent to the curve because it emerges from the limiting direction of secant vectors as they shrink to a point. As the time interval h approaches zero, the secant vector's direction aligns with the curve's direction at that point, making the derivative vector tangent to the path.

Scaling the secant vector by 1/h preserves its direction but adjusts its magnitude proportionally. This operation converts displacement into a rate of change, transforming the secant vector into the difference quotient that approaches instantaneous velocity as the time interval h approaches zero.