Derivative of the Dot Product

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

In other words, the derivative of the dot product expands into two terms:

• The dot product of the derivative of the first vector (a') with the second vector (b), which remains unchanged.
• The dot product of the first vector (a), unchanged, with the derivative of the second vector (b').

This rule generalizes naturally to more complex expressions involving time-dependent vectors.