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Delicate Derivative Derivation II (Posted on 20100803) 

F(x) is a function which is defined as:
F(x) = √(1 + 4x^{2}  4x^{3}  4x^{5} + 4x^{6} + x^{8})
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)


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) = a_{0} + a_{1}x + a_{2}x^{2} + ... then
F^{(10)}(x) = 10!a_{10} + 11!a_{11}x + (12!/2!)a_{12}x^{2} + ... giving F^{(10)}(0) = 10!a_{10}
So we only need to find a_{10}, the coefficient of x^{10}, and neglect all other terms.
Let F(x) = sqrt(1 + x^{2}f) where f = 4  4x  4x^{3} + 4x^{4} + x^{6}
F(x) = 1 + (1/2)x^{2}f  (1/8)x^{4}f^{2} + (1/16)x^{6}f^{3}  (5/128)x^{8}f^{4} + (7/256)x^{10}f^{5} + ...
To find all the appearances of x^{10} we need to consider each part in turn:
1 + (1/2)x^{2}f contains no term in x^{10}
x^{4}f^{2} contains the terms: x^{4} [2(4)(x^{6}) + (4x^{3})^{2}] = 24x^{10}
x^{6}f^{3} contains the terms: x^{6} [3(4)^{2}(4x^{4}) + 6(4)(4x)(4x^{3})] = 576x^{10}
x^{8}f^{4} contains the term: x^{8} 6(4)^{2}(4x)^{2} = 1536x^{10}
x^{10}f^{5} contains the term: x^{10} (4)^{5} = 1024x^{10}
all later terms do not contain x^{10}
Collecting these together gives the complete term in x^{10} as:
 (1/8)24x^{10} + (1/16)576x^{10}  (5/128)1536x^{10} + (7/256)1024x^{10}
= (3 + 36  60 + 28)x^{10}
= x^{10}
Therefore, a_{10} = 1 and F^{(10)}(0) = 10! = 3628800

Posted by Harry
on 20100805 00:36:32 


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