Part 5/5:
This result is fundamental in vector calculus, often used in deriving equations of motion, electromagnetic theory, and many other applications involving vector fields. It generalizes the product rule into the vector domain and emphasizes the importance of component-wise differentiation and summation.
Conclusion
By expressing the dot product as a summation and applying the basic product rule component-wise, the proof demonstrates that the derivative of the dot product of vector functions follows a natural extension of the scalar product rule. This foundational formula enables mathematicians and physicists to efficiently handle derivatives of vector expressions in various advanced contexts.