Mathematical Structure
Differentiating Vector-Valued Functions

Prerequisites:

The derivative of a vector-valued function

Missing equation

is defined in exactly the same way as the derivative of an ordinary function.

Missing equation

The argument below shows that the derivative of a vector-valued function with values in R^n

Missing equation

can be found by differentiating each of the coordinates.

Missing equation

The CAS files below show how each computer algbera system can differentiate vector-valued functions.

Maple worksheet Mathematica notebook TI-92 Browser Window

Help Help Help

The formula for differentiating a dot product of vector-valued functions is very similar to the formula for differentiating an ordinary product of ordinary functions

Theorem

If

Missing equation

Then

Missing equation

Proof:

Missing equation

Find the derivative of each of the following functions.

  1. F(t) = (sin t, cos t, 3 t)

  2. F(t) = (2 t + 2, 3 t + 4, 4 t + 5)

  3. F(t) = (1 - t2, 2 t3, 3 t)

Copyright c 1995 by Frank Wattenberg, Department of Mathematics, Montana State University, Bozeman, MT 59717