WHAT WAS THE QUESTION?            DERIVE INVESTIGATIONAL ACTIVITY

Differentiating Products

You already know how to differentiate some basic functions:

D( xn ) = n xn-1

D( eax ) = a eax

D( sin ax ) = a cos ax

D( ln x ) = 1 / x

D( cos ax ) = - a sin ax

We are now going to find a rule for differentiating more complicated functions, which are products of the basic functions listed above.

Examples of such products are: x2 . sin x          e4x . cos 3x          5x . e-x         10x7 . ln x

 

EXERCISE 1

Use DERIVE to differentiate these functions:

1) D( x3 . sin x ) =

2) D( x2 . cos x ) =

3) D( sin x . ln x ) =

4) D( 5x . sin 2x ) =

5) D( e4x . cos x ) =

6) D( e-x . sin 4x ) =

Now write down what you think you get when you differentiate the functions below.
Then use DERIVE to check your answers, and correct them if necessary:

7) D( x4 . sin x ) =

8) D( x6 . cos x ) =

9) D( e-3x . sin x ) =

10) D( sin 2x . ln x ) =

11) D( 4x3 . cos 5x ) =

12) D( ex . sin 2x ) =

 

EXERCISE 2 (What was the question?)

Differentiate these functions:

1) D(                ) =     x6 . cos x     +     6x5 . sin x

2) D(                ) =     8x . cos x     -     4x2 . sin x

3) D(                ) =     sin 3x / x     -     3 cos 3x . ln x

4) D(                ) =     24x2 . cos 3x     +    16x . sin 3x

5) D(                ) =     x5 . ex     +     5x4 . ex

6) D(                ) =     8 e2x . sin 2x     +     8 e2x . cos 2x

 

EXERCISE 3

If you were given a product U . V , where U and V are basic functions of x, explain how you would find D( U . V )

 

 

Hints on using DERIVE

 

For Exercise 1

To differentiate a function of x, do the following:

<A>uthor . . . the function . . . . <enter>

<C>alculus

<D>ifferentiate

<enter> to confirm line number

<enter> to confirm variable is x

<enter> to confirm order 1 (meaning only differentiate once)

<S>implify

 

For Exercise 2

In Exercise 1, you should have spotted the kind of thing that happens when you differentiate a product. Use any patterns you noticed to guess what the original function would have been in Exercise 2. Use DERIVE to differentiate your guess as described above. Do you get the answer that is expected? If not, modify your guess and try again.

NOTE: It does not matter which order things are written in, so long as everything is there.

For example, 4 cos x . e2x is the same as 4 e2x . cos x

 

David Bowers,
Mathematics Workshop,
Suffolk College.