Definition of Derivative for Vector Functions

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2025-09-16 04:44

In this video I go over the definition of derivative for a vector function, which is very similar to the definition for real valued functions. The derivative of a vector function is defined as the limit as h approaches zero of the secant vector (difference of two vectors separated by incrementing the parameter t by h) divided h. As h approaches zero the secant vector approaches a vector that is parallel to the tangent vector to the curve. I illustrate this with an animation using GeoGebra. I also quickly define the unit tangent vector, which is just the tangent vector of length 1 (obtained by dividing the tangent vector by its magnitude).

#math #science #calculus #vectorfunctions #geogebra

1 Derivatives of Vector Functions.jpeg

Timestamps

Introduction to Derivatives and Integrals of Vector Functions: 0:00
Definition of Derivative of a Vector Function: 0:15
Secant vector is the vector between the position vectors in the definition of derivative: 1:36
Tangent vector is secant vector multiplied by 1/h as h approaches zero: 4:28
GeoGebra animation of the secant line approaching the tangent line: https://www.geogebra.org/m/sVCRDDmA 7:17
Unit Tangent Vector is the tangent vector divided by its magnitude: 10:16