Show that (i) \(\mathrm{D}(\boldsymbol{a} \wedge \boldsymbol{b})=(\mathrm{D} \boldsymbol{a}) \wedge \boldsymbol{b}+\boldsymbol{a} \wedge \mathrm{D} \boldsymbol{b}\), (ii) \(\frac{\mathrm{d}}{\mathrm{d} t}(\boldsymbol{a} \cdot \boldsymbol{b})=(\mathrm{D} \boldsymbol{a}) \cdot \boldsymbol{b}+\boldsymbol{a} \cdot(\mathrm{D} \boldsymbol{b})\).

The derivative operator, often denoted by symbols like \( \mathrm{D} \) or \( \frac{\mathrm{d}}{\mathrm{d} t} \), is a fundamental tool in calculus that measures the rate at which a quantity changes. In essence, it provides a way to calculate the instantaneous rate of change of a function with respect to one of its variables. For a function \( f(x) \), the derivative \( f'(x) \) or \( \frac{\mathrm{d}f}{\mathrm{d}x} \) represents the slope of the tangent line to the function's graph at any point \( x \).

When we apply the derivative operator to vector functions, the operation is conducted element-wise. This means that if \( \boldsymbol{a}(t) \) and \( \boldsymbol{b}(t) \) are vector functions, their derivatives \( \mathrm{D} \boldsymbol{a} \) and \( \mathrm{D} \boldsymbol{b} \) are vectors where each component is the derivative of the corresponding component of the original vectors. This concept is crucial for understanding many physical phenomena, particularly in fields that involve motion and forces, such as mechanics and electromagnetism.

It is this operator that we use to apply the product rule for differentiation in vector calculus, which allows us to handle the differentiation of products of vector functions systematically, rather than component by component.

Vector functions are functions that have vectors as their values, rather than scalars. In other words, each input to a vector function corresponds to a vector in a space, typically in \( \mathbb{R}^n \) (n-dimensional real coordinate space). An example of a vector function is \( \boldsymbol{a}(t) \), where the input \( t \) could represent time and the output is a vector, indicating perhaps the position of a particle at time \( t \).

The differentiation of vector functions follows the same principles as scalar functions, but is done component-wise. When vector functions interact—through operations like the dot product or the cross product—differentiating these interactions correctly often requires a rule known as the product rule.

The product rule is a fundamental principle for successfully dealing with derivatives of products of vector functions. It's crucial for students to understand that when vector functions are multiplied (whether through dot or cross product), and we want to take the derivative of this product, the rule ensures we account for the change in both vector functions independently, similar to how we'd approach the derivative of a product of two scalar functions.

The time derivative is a specific case of the derivative operator that measures how a quantity changes over time. Symbolically represented as \( \frac{\mathrm{d}}{\mathrm{d}t} \), it specifically looks at the rate of change with respect to time. This particular derivative is essential in physics and engineering, where change with time is a fundamental aspect of most problems.

When we speak of the time derivative of vector functions, we are looking at how these vectors change as time progresses. For motion, this could mean finding the velocity (the time derivative of position) or acceleration (the time derivative of velocity). For a vector function like \( \boldsymbol{a}(t) \) where \( t \) stands for time, the time derivative \( \frac{\mathrm{d}\boldsymbol{a}}{\mathrm{d}t} \) gives us a new vector function that shows the instantaneous rate of change of \( \boldsymbol{a} \) with respect to time at each moment.

In practical terms, understanding and applying the time derivative allows one to predict future states of dynamic systems, such as the trajectory of a projectile, the oscillations of a pendulum, or the flow of electricity in a circuit. It is the starting point for formulating many laws of nature, like Newton's second law of motion, which fundamentally couples force to the time derivative of momentum.