Vector Derivative — Definition, Formula & Examples

A vector derivative is the derivative of a vector-valued function, computed by differentiating each component of the vector separately with respect to the parameter (usually time).

Given a vector-valued function $\mathbf{r}(t) = \langle f(t),\, g(t),\, h(t) \rangle$ , its derivative is defined as $\mathbf{r}'(t) = \lim_{\Delta t \to 0} \frac{\mathbf{r}(t + \Delta t) - \mathbf{r}(t)}{\Delta t}$ , which equals $\langle f'(t),\, g'(t),\, h'(t) \rangle$ provided each component function is differentiable at $t$ .

Key Formula

\mathbf{r}'(t) = \left\langle \frac{df}{dt},\, \frac{dg}{dt},\, \frac{dh}{dt} \right\rangle

Where:

$\mathbf{r}(t)$ = A vector-valued function of the parameter t
$f(t),\, g(t),\, h(t)$ = The scalar component functions of the vector
$t$ = The parameter, often representing time

How It Works

To differentiate a vector-valued function, apply the standard rules of single-variable calculus to each component independently. The resulting derivative vector

\mathbf{r}'(t)

is tangent to the curve traced by

\mathbf{r}(t)

at the point corresponding to parameter

t

. Its magnitude

\|\mathbf{r}'(t)\|

gives the speed of a particle moving along that curve. Product rules for dot products and cross products carry over with modifications: for instance,

\frac{d}{dt}[\mathbf{u}(t) \cdot \mathbf{v}(t)] = \mathbf{u}'(t) \cdot \mathbf{v}(t) + \mathbf{u}(t) \cdot \mathbf{v}'(t)

Worked Example

Problem: Find the derivative and speed at t = 1 for the vector-valued function r(t) = ⟨t², 3t, sin(πt)⟩.

Differentiate each component: Apply standard differentiation rules to each component separately.

\mathbf{r}'(t) = \langle 2t,\, 3,\, \pi\cos(\pi t) \rangle

Evaluate at t = 1: Substitute t = 1 into each component of the derivative.

\mathbf{r}'(1) = \langle 2,\, 3,\, \pi\cos(\pi) \rangle = \langle 2,\, 3,\, -\pi \rangle

Find the speed: The speed is the magnitude of the derivative vector.

\|\mathbf{r}'(1)\| = \sqrt{4 + 9 + \pi^2} = \sqrt{13 + \pi^2} \approx 4.79

Answer: The derivative at t = 1 is ⟨2, 3, −π⟩, and the speed is √(13 + π²) ≈ 4.79.

Why It Matters

Vector derivatives are essential in physics for describing velocity and acceleration of objects moving through space. In multivariable calculus and differential geometry, they underpin the computation of tangent vectors, arc length, and curvature of space curves.

Common Mistakes

Mistake: Forgetting that the derivative of a vector-valued function is itself a vector, and treating the result as a scalar.

Correction: Always write the derivative as a vector with the same number of components as the original function. Each component is differentiated independently.