Derivatives of Vector Functions

Calculus 3 · Axiom Academy

LESSON Derivatives of Vector Functions Componentwise differentiation, the tangent vector, and the geometric meaning of a derivative in space. 1. Componentwise Differentiation Each component is differentiated independently, using the same rules from single-variable calculus. The result is a new vector function that represents the rate of change at each point along the curve. The helix we will follow for the rest of the lesson If , then differentiating slot by slot gives . No new machinery — three ordinary derivatives, kept in their places. 2. Geometric Interpretation: The Tangent Vector The derivative is not just a mathematical operation — it has profound geometric meaning. It gives us a vector that points in the direction the curve is heading at time t . The numerator is the chord joining two points of the curve. Dividing by h rescales that chord into the secant vector . As h shrinks, the chord collapses to nothing — but the rescaled secant vector does not: it pivots until it lies along the curve. That limiting arrow is . On the helix take , where . The difference quotient is The third slot is for every h — which is why the arrow's climb never changes while its horizontal part swings into place. 3. Velocity and Speed Along the Curve Read as the position of a moving particle and becomes its velocity : a single arrow that carries both pieces of information at once. The arrow points along the curve, in the direction of increasing t — it is tangent at every instant.

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