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

Graphical or geometric meaning

At $t=0$, nearby secant arrows approach the tangent direction $\langle0,1,1\rangle$. The velocity lies in three-dimensional space, while speed is the nonnegative scalar length of that arrow.

Common mistakes and why they fail

Verification and reasonableness checks

• Use componentwise difference quotients for a sample component.
• Verify speed is nonnegative and has distance-per-parameter units.
• Check tangent point and direction separately in a line equation.

Differentiate components, interpret the vector

For $\mathbf r(t)=\langle x(t),y(t),z(t)\rangle$, differentiation is componentwise. The result $\mathbf r'(t)$ is velocity and is tangent when nonzero; its magnitude is speed. The second derivative is acceleration, whose decomposition into tangential and normal components separates change of speed from change of direction. Product rules remain valid for dot and cross products, but cross-product order matters. A constant norm implies $\mathbf r\cdot\mathbf r'=0$, so motion on a sphere has velocity perpendicular to the radius; this gives a useful geometric check. When evaluating at a parameter value, distinguish the point $\mathbf r(t)$ from its tangent vector $\mathbf r'(t)$ and attach the appropriate physical units.

Practice

1. Differentiate $\langle t^3,\cos t\rangle$.
2. Find speed of $\langle3t,4t\rangle$.
3. If $\|\mathbf r\|$ is constant, what is $\mathbf r\cdot\mathbf r'$?

Connections and next steps

• Vector-Valued Functions
• Space Curves
• Arc Length and Curvature