We think about the derivative of vector-valued functions.

**Work in groups of 3–4, writing your answers on a separate sheet of
paper.**

Previously in calculus course you learned the following metaphor for the derivatives:

Given a function , the derivative of is the slope of the tangent line at any point on the graph .

This is really a great metaphor for functions that map from to . However, now we are studying vector-valued functions. A new metaphor is needed:

Given a vector-valued function , the derivative of is a tangent vector at any point on the graph of .

Let’s see if we can figure what this is saying.

### Lines

Suppose you have a line given by the vector valued function .

### Circles

The a circle of radius centered at is given by and here is a plot:

### Projectile motion

Vector-valued functions are excellent for modeling projectile motion. The function below models the path of a calculus book being thrown from an initial height of at an initial velocity of at a angle: For your viewing pleasure here is a plot:

### The moral of the story

The moral of the story is this: When studying, functions from to , it makes a lot of sense to plot their derivatives. When dealing with vector-valued functions, plotting their derivatives might not be the best idea. Instead you should be plotting tangent vectors.