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Delicate Derivative Derivation II (Posted on 2010-08-03) Difficulty: 3 of 5
F(x) is a function which is defined as:
F(x) = √(1 + 4x2 - 4x3 - 4x5 + 4x6 + x8)

Determine F(10)(0), that is, the tenth derivative of F(x) with respect to x at x=0.

No Solution Yet Submitted by K Sengupta    
Rating: 1.0000 (1 votes)

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Solution Solution | Comment 3 of 4 |
K.S. seems to have chosen the function F(x) very carefully, so there’s probably a much more subtle approach than my bludgeoning attack shown below.

If F(x) is expanded as a power series:  F(x) = a0 + a1x + a2x2 + ...     then

F(10)(x) = 10!a10 + 11!a11x + (12!/2!)a12x2 + ...    giving       F(10)(0) = 10!a10

So we only need to find a10, the coefficient of x10, and neglect all other terms.

Let  F(x) = sqrt(1 + x2f)              where f = 4 - 4x - 4x3 + 4x4 + x6

F(x) = 1 + (1/2)x2f - (1/8)x4f2 + (1/16)x6f3 - (5/128)x8f4 + (7/256)x10f5 + ...

To find all the appearances of x10 we need to consider each part in turn:

     1 + (1/2)x2f  contains no term in x10

     x4f2 contains the terms:          x4 [2(4)(x6) + (-4x3)2] = 24x10

     x6f3 contains the terms:          x6 [3(4)2(4x4) + 6(4)(-4x)(-4x3)] = 576x10

     x8f4 contains the term:           x8 6(4)2(-4x)2 = 1536x10

     x10f5 contains the term:          x10 (4)5 = 1024x10

     all later terms do not contain x10

Collecting these together gives the complete term in x10 as:

            - (1/8)24x10 + (1/16)576x10 - (5/128)1536x10 + (7/256)1024x10

=          (-3 + 36 - 60 + 28)x10

=          x10

Therefore,   a10 = 1   and   F(10)(0) = 10!  =  3628800



  Posted by Harry on 2010-08-05 00:36:32
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