Derivative of the Cross Product

The derivative of the cross product of two vectors is expressed as the sum of two cross products: the cross product of the derivative of the first vector with the second vector, plus the cross product of the first vector with the derivative of the second vector. $$ \frac{d \ [ \vec{a} \ × \ \vec{b} ]}{dt} = \frac{d \ \vec{a} }{dt} \ × \ \vec{b} \ + \ \vec{a} \ × \ \frac{d \ \vec{b} }{dt}$$

More precisely, the derivative of $ a \times b $ expands as follows:

• The cross product of the derivative of the first vector (a') with the second vector (b), held constant.
• The cross product of the first vector (a), held constant, with the derivative of the second vector (b').

This result follows directly from the product rule and applies broadly to time-dependent vectors in dynamic systems.