So I was wondering about the interpretation of the derivative of a vector-valued function in the sense of how it is analogous to a function with one variable input and one output. With the derivative in single variable calculus, it is interpreted as the slope of the line that is tangent to the curve.
With a vector valued function, when we take the derivative, it is equivalent to taking the derivative of each of the components of the function's output. So are we saying that instead of a representing a line that is tangent to the curve, it is representing a vector that is tangent to the curve?
But what is confusing me about the analogy is this. In the single variable situation, we can travel along the tangent line that is given by the derivative. What is the meaning of "travelling along the tangent line" when we have a tangent vector? Do we travel along the tangent vector?
If $f: \mathbb R \to \mathbb R^n$ is differentiable, then $\{ f(t) \mid t \in \mathbb R \}$ is a curve in $\mathbb R^n$. The tangent line to this curve at a point $f(t_0)$ consists of all points in $\mathbb R^n$ of the form $f(t_0) + f'(t_0)(t - t_0)$ where $t \in \mathbb R$. You can visualize that tangent line.