Let's do some calculus! (35)

Calculus Level 5

Let f(x)=cos(cos(cos(cos(cos(cos(cos(cosx)))))))f(x) = \cos \left( \cos \left( \cos \left( \cos \left( \cos \left( \cos \left( \cos \left( \cos x \right) \right) \right) \right) \right) \right) \right) , and suppose that the number jj satisfies the equation j=cosjj = \cos j. If f(j)f'(j) can be expressed as a polynomial in jj as

f(j)=aj8+bj6+cj4+dj2+ef'(j) = a j^8 + b j^6 + c j^4 + d j^2 + e

where a, b, c, da, \ b, \ c, \ d and ee are integers. Then find the value of a+b+c+d+e |a| + |b| + |c| + |d| + |e|.

Notation: f()f'(\cdot) denotes the first derivative of the function f()f(\cdot).

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