Derivatives of Vector Functions

The central idea

For $\mathbf r:I\to\mathbb R^n$, differentiation is componentwise: $\mathbf r'(t)=\langle r_1'(t),\ldots,r_n'(t)\rangle$. In motion, $\mathbf r'$ is velocity, $\|\mathbf r'\|$ is speed, and $\mathbf r''$ is acceleration. Product rules include $(\mathbf u\cdot\mathbf v)'=\mathbf u'\cdot\mathbf v+\mathbf u\cdot\mathbf v'$.

Definitions, hypotheses, and notation

Differentiating $\mathbf r\cdot\mathbf r=R^2$ for motion on a sphere gives $2\mathbf r\cdot\mathbf r'=0$, so velocity is tangent and perpendicular to the radius. This is a useful geometric consequence of the dot-product rule. The cross-product rule is $(\mathbf u\times\mathbf v)'=\mathbf u'\times\mathbf v+\mathbf u\times\mathbf v'$; factor order matters.

Higher derivatives remain vectors in the same ambient space. A zero velocity can make the usual tangent formula fail even when the path has a limiting tangent, so regularity should be checked rather than assumed from a smooth component formula.

Conceptual meaning

The derivative vector is the limit of secant displacement vectors divided by parameter change. It is tangent to a regular curve and points in the direction of increasing parameter. Its magnitude depends on traversal speed, not only geometric shape.

A dependable method and decision rule

1. Differentiate every component with respect to the same parameter.
2. Keep vector, scalar, dot, and cross-product output types distinct.
3. Evaluate at the requested parameter value.
4. Take a norm only when speed or magnitude is requested.
5. Check regularity before using the derivative as a tangent direction.

Fully worked example

Common mistakes and why they fail