Part 1/5:

Proof of the Derivative Formula for the Dot Product of Vector Functions

Understanding how to differentiate the dot product of two vector functions is fundamental in vector calculus. This concept extends the familiar product rule from basic calculus to vector functions, and the process involves careful application of rules for multiple components and summations. Here, we explore this proof thoroughly, starting from defining the vector functions, expressing their dot product, and ultimately deriving the formula for their derivative.

Defining the Vector Functions

Let’s consider two vector functions, ( \mathbf{u}(t) ) and ( \mathbf{v}(t) ), each with three components:

[

\mathbf{u}(t) = \begin{bmatrix}

f_1(t) \

f_2(t) \

f_3(t)

\end{bmatrix}

\quad \text{and} \quad