State and prove the product rule for the dot product of 2 vector valued functions. You may assume the distributive law.

The product rule for the dot product of two vector-valued functions states that:

\[
\frac{d}{dt}[\mathbf{u}(t) \cdot \mathbf{v}(t)] = \mathbf{u}'(t) \cdot \mathbf{v}(t) + \mathbf{u}(t) \cdot \mathbf{v}'(t)
\]

### Proof:
Let \(\mathbf{u}(t)\) and \(\mathbf{v}(t)\) be vector-valued functions of \(t\), both differentiable. We aim to differentiate their dot product with respect to \(t\).

#### Step 1: Write the dot product as a scalar function
The dot product of two vectors \(\mathbf{u}(t)\) and \(\mathbf{v}(t)\) is a scalar function:
\[
\mathbf{u}(t) \cdot \mathbf{v}(t) = \sum_{i=1}^{n} u_i(t)v_i(t)
\]
where \(u_i(t)\) and \(v_i(t)\) are the components of \(\mathbf{u}(t)\) and \(\mathbf{v}(t)\), respectively.

#### Step 2: Differentiate the scalar dot product
By differentiating the dot product component-wise using the product rule for scalar functions, we get:
\[
\frac{d}{dt}[\mathbf{u}(t) \cdot \mathbf{v}(t)] = \frac{d}{dt} \left( \sum_{i=1}^{n} u_i(t)v_i(t) \right)
\]

Using the product rule for scalar differentiation:
\[
\frac{d}{dt} \left( u_i(t)v_i(t) \right) = u_i'(t)v_i(t) + u_i(t)v_i'(t)
\]

So, summing over all components:
\[
\frac{d}{dt}[\mathbf{u}(t) \cdot \mathbf{v}(t)] = \sum_{i=1}^{n} \left( u_i'(t)v_i(t) + u_i(t)v_i'(t) \right)
\]

#### Step 3: Re-express in vector notation
We can now combine the sums back into vector form:
\[
\frac{d}{dt}[\mathbf{u}(t) \cdot \mathbf{v}(t)] = \mathbf{u}'(t) \cdot \mathbf{v}(t) + \mathbf{u}(t) \cdot \mathbf{v}'(t)
\]

Thus, the product rule for the dot product is proven.

### Conclusion:
The derivative of the dot product of two differentiable vector-valued functions is given by the sum of the dot product of the derivative of the first function with the second, and the dot product of the first function with the derivative of the second.

### Related Questions:
1. How is the product rule for scalar functions derived?
2. Can the product rule be extended to cross products?
3. What are the implications of the product rule in physics?
4. How does the product rule differ when dealing with matrix-valued functions?
5. What happens to the product rule when the vector functions are orthogonal?

### Tip:
The product rule for the dot product is frequently used in applications involving motion, where vectors represent velocity and acceleration.

State and prove the product rule for the dot product of two vector-valued functions. You may assume the distributive law.

Explore the product rule for the dot product of two vector-valued functions with a complete proof. This detailed explanation covers key concepts in vector calculus, including differentiation, vector functions, and the distributive law. Suitable for undergraduate students studying calculus.

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