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.

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