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. ✨ 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 = lim Δu→0 [R(u+Δu) − R(u)] / Δu = lim Δu→0 [x(u+Δu)î + y(u+Δu)ĵ + z(u+Δu)k̂ − (x(u)î + y(u)ĵ + z(u)k̂)] / Δu = lim Δu→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 t...