Let's do some calculus! (35)

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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