Derivatives of Vector Functions

A vector-valued function describes position as a function of time. For example, in Cartesian coordinates, the position of a car moving along a curved road can be written as

\begin{equation*}\textbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}

Secant Vector and Average Velocity:

This secant vector captures the overall change in position during the interval and provides a crude estimate of the direction of motion.

At time t, the car is at point P, with position r(t). After a short interval h, it reaches point Q, with position r(t+h). The secant vector from P to Q is

\begin{equation*}\textbf{r}(t+h)-\textbf{r}(t)\end{equation*}

\begin{equation*}=\langle x(t+h),y(t+h),z(t+h)\rangle - \langle x(t),y(t),z(t)\rangle\end{equation*}

\begin{equation*}=\langle x(t+h)-x(t),y(t+h)-y(t),z(t+h)-z(t)\rangle\end{equation*}

To get the average velocity, the secant vector is scaled by the reciprocal of the time interval, forming the difference quotient:

\begin{equation*}\jfrac{\textbf{r}(t+h)-\textbf{r}(t)}{h} = \left\langle \jfrac{x(t+h)-x(t)}{h}, \jfrac{y(t+h)-y(t)}{h}, \jfrac{z(t+h)-z(t)}{h}\right\rangle\end{equation*}

While the direction of this vector remains consistent with the secant vector, its magnitude now reflects the average speed over the interval h.

Derivative Vector:

As h approaches zero, point Q converges to point P, and the secant vector becomes infinitesimally short. The limit of the difference quotient as h→0 yields the derivative vector:

\begin{equation*}\textbf{r}'(t)=\lim_{h\to 0}\jfrac{\textbf{r}(t+h)-\textbf{r}(t)}{h}\end{equation*}

The limit can be evaluated component-wise:

\begin{equation*}\textbf{r}'(t)= \left\langle \lim_{h\to 0}\jfrac{x(t+h)-x(t)}{h}, \lim_{h\to 0}\jfrac{y(t+h)-y(t)}{h}, \lim_{h\to 0}\jfrac{z(t+h)-z(t)}{h} \right\rangle\end{equation*}

\begin{equation*}\textbf{r}'(t)=\langle x'(t),y'(t),z'(t)\rangle\end{equation*}

This derivative vector is tangent to the curve at point P and represents the car’s instantaneous velocity, including both speed and direction.