Derivative Rules

     Many of the following derivative rules follow from the same techniques as when proving their one variable counterparts (along with several vector properties).

Proposition (Derivative Rules for Vector Functions) If the vector functions and the scalar function are differentiable at , and if and are constants, then are differentiable at and,

    (i)  Linearity:  
    
    (ii)  Scalar Multiple:
    
    (iii)  Dot Product:
    
    (iv)  Cross Product:
    
    (v)  Chain Rule:
        
    Proof. The linearity rule follows from:









The scalar multiple rule follows from:









The dot product rule (and similar for the cross product rule) follows from:











The rest of the proof is left as an exercise for the reader.

Example (Derivative Rules for Vector Functions) Compute the derivative of the vector function given by where

    and     
    
    Solution. We have,

Proposition (Vector Function and Derivative) If the nonzero vector function is differentiable and has constant length, then is orthogonal to the derivative vector

    Proof. The vector function has constant length means there is a real number, such that for all . We take the derivative of both sides obtaining, Thus, and so and are orthogonal.

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Derivative Rules
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/derivative-rules.html