Let's do some calculus! (35)

Calculus Level 4

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

\[f'(j) = a j^8 + b j^6 + c j^4 + d j^2 + e\]

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

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

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