Derivative of a Product Let g ( x) = x 7 and let h( x) = x11 Let f ( x) = g ( x ) h( x) 1. Find g ( x) 2. Find h( x)

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3. Write an equation for f ( x ) as a single power of x. Then find an equation for f ( x)

4. Show that f ( x) does not equal g ( x) h( x)

5. It is possible to get the correct answer for f ( x) by a clever combination of the equations for g ( x), h( x) , g ( x) , and h( x) . For instance, you might notice that the 18 in 18x17 is the sum of the 7 and 11 in the 7x 6 and 11x10 . Figure out what this combination is.

6. Make a conjecture about what f ( x) equals in terms of g ( x), h( x) , g ( x) , and h( x) .

7. Assume that your conjecture in question 6 is true for any product of two functions. If f ( x) = ( x 4 )(5 x 2 ) , what would f ( x) equal?

Find f ( x) . 1. f ( x) = ( x + 1)(3 x 4)

2. f ( x) = x (2 x + 1)

3. f ( x) = (2 x 2 3 x + 5)( 3 x2 2)

4. f ( x) = (3 x 2 4)(7 x8 + 5 x4 3x3 2 x + 1100)