91Ó°ÊÓ

When working with vector-valued functions, finding the derivative involves determining the rate at which the entire function changes with respect to a variable, usually time \( t \). A vector-valued function can be thought of as a 'path' traced by continuous motion in space, formed by its component functions.

• Each component of the vector function is differentiated separately, as if it were an ordinary scalar function of \( t \).
• The differentiation process is carried out independently for each of the basis vectors like \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).
• The resulting derivative is another vector-valued function, that shows the rate of change along each axis.

Considering instance the vector function \( \mathbf{r}(t) = t^3 \mathbf{i} + 3t^2 \mathbf{j} + \frac{t^3}{6} \mathbf{k} \), to find \( \mathbf{r}'(t) \), each term is differentiated using standard calculus rules. By differentiating each component function, you can create the complete derivative vector.

The power rule is a fundamental tool in calculus for finding derivatives of functions that are powers of \( t \), such as \( t^n \). This rule states that the derivative of \( t^n \) with respect to \( t \) is given by \( \frac{d}{dt}(t^n) = nt^{n-1} \).

• This rule is applicable to each individual component in a vector-valued function.
• It simplifies the process, as long as the function is a single term power of \( t \).
• Even in cases where there are coefficients, such as having \( 3t^2 \) or \( \frac{1}{6}t^3 \), you simply multiply the derivative by the coefficient.

For example, in the function \( \mathbf{r}(t) = t^3 \mathbf{i} + 3t^2 \mathbf{j} + \frac{t^3}{6} \mathbf{k} \), utilizing the power rule on \( t^3 \) yields \( 3t^2 \), applying it to \( 3t^2 \) results in \( 6t \), and for \( \frac{1}{6}t^3 \) it results in \( \frac{1}{2}t^2 \). This efficient rule streamlines computing derivatives in mathematics.

A vector in mathematics is composed of multiple components, typically represented with respect to standard basis vectors such as \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). These basis vectors point along the mutually perpendicular x, y, and z axes in three-dimensional space:

• The \( \mathbf{i} \)-component aligns with the x-axis.
• The \( \mathbf{j} \)-component aligns with the y-axis.
• The \( \mathbf{k} \)-component aligns with the z-axis.

Each component of a vector-valued function can be an expression involving different terms of \( t \). In the function \( \mathbf{r}(t) = t^3 \mathbf{i} + 3t^2 \mathbf{j} + \frac{t^3}{6} \mathbf{k} \), the \( t^3 \) represents movement along the x-axis, \( 3t^2 \) along the y-axis, and \( \frac{t^3}{6} \) along the z-axis.

• Understanding individual components is key for breaking down complex vector functions into manageable parts.
• It helps isolate each part of the vector equation, making the process of finding derivatives clearer and more straightforward.

Through separate differentiation of each component, it guarantees that changes in direction and magnitude are accurately represented in the derivative vector.