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Differentiating Vector-Valued Functions

So far, we've been dealing with functions that output a single value. Now, we consider functions that output a vector.

Consider the following parametric function:

This can be written as a vector-valued function:

Table of Contents​

Deriving the Parametric Derivative​

Vector-valued functions, like many functions, can be thought of as a transformation. In this case, it transforms a point in a number line to a point in a 2D plane.

Consider taking the derivative of this function. Recall that by taking the derivative, you increase the input by a small amount, and see how the output changes. A drawing of a parametric function can help illustrate this:

The change in the input is and the change in the output is .

Therefore, the derivative can be written as:

Since , the derivative can be computed as:

This is essentially the same as taking the derivative of a function that outputs a single value, but for each component of the vector.

Another way to think about this is to consider the change in visually:

Since , as , , and:

This can be thought of as the implicit differentiation of vector-valued functions.

The Magnitude of the Parametric Derivative​

The magnitude of the parametric derivative is the rate of change of the vector-valued function. To illustrate this, consider two different vector-valued functions:

First, let's graph :

Notice that, at , the curve goes halfway around the unit circle. Now consider :

In this case, at , the curve goes halfway around the unit circle. So it's twice as fast as , if you consider as physical time.

Let's consider the derivatives of these functions:

Let's evaluate the derivatives at certain points and plot them:

The important takeaway is that even if the slope appears the same, the rate of change of the vector-valued function can be different.

Summary and Next Steps​

In this section, we introduced the concept of differentiating vector-valued functions.

Here are the key points to remember:

  • The derivative of a vector-valued function is the rate of change of the function.
  • The derivative of a vector-valued function is also a vector.
  • The derivative of a vector-valued function can be thought of as the implicit differentiation of the components of the function.

In the next section, we will introduce the multivariable chain rule, which is used to differentiate composite functions of multiple variables.