Char. Limit said:That doesn't make too much sense if r(t) is supposed to be constant along the j unit vector. My best guess is that it's probably a misprint, and that r(t) is supposed to be r(t)=ln(t) i + 1/t j. That said, if this is for an assignment, proceed with the problem as written, which you're on the right track so far.
A vector function is a mathematical function that takes an input or inputs and maps them to a vector as an output. It can be thought of as a vector-valued function.
The derivative of a vector function is a vector that describes the rate of change of the function with respect to its input parameters. It measures how much the vector function changes for a small change in the input parameters.
The derivative of a vector function is calculated by taking the partial derivatives of each component of the vector function with respect to its input parameters. These partial derivatives are then combined to form a new vector, which is the derivative of the original vector function.
The derivative of a vector function is significant because it helps us understand the behavior of the vector function at a particular point. It can be used to determine the slope or direction of the vector function at that point, as well as the rate of change of the function in different directions.
The derivative of a vector function has many practical applications in fields such as physics, engineering, and economics. It is used to model and analyze systems that involve vector quantities, such as velocity, acceleration, and force. It is also used in optimization problems to find the maximum or minimum value of a vector function.