How to Derive the Derivative of a Vector Function

How to Derive the Derivative of a Vector Function 🧮

Let’s start with a vector function of a single variable:

R(u) = x(u)î + y(u)ĵ + z(u)k̂

Here, x, y, and z are differentiable scalar functions of a real parameter u. This means that as u changes, the point R(u) moves through space — tracing a smooth curve. ✨

A smart student thinking about calculus.

The Goal

We want to find dR/du — the rate at which the vector R(u) changes with respect to u.

Proof

By definition of the derivative:

dR/du = limΔu→0 [R(u+Δu) − R(u)] / Δu

= limΔu→0 [x(u+Δu)î + y(u+Δu)ĵ + z(u+Δu)k̂ − (x(u)î + y(u)ĵ + z(u)k̂)] / Δu

= limΔu→0 [(x(u+Δu)−x(u))/Δu]î + [(y(u+Δu)−y(u))/Δu]ĵ + [(z(u+Δu)−z(u))/Δu]k̂

= (dx/du)î + (dy/du)ĵ + (dz/du)k̂

Interpretation

The derivative dR/du is itself a vector — one that points in the direction of motion of R(u) and whose magnitude gives the speed of change. Each component (x, y, z) behaves just like an ordinary function, so we can differentiate them individually and recombine them to form the derivative vector. 🎯

* For educational use only

Date: 01.11.2025

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