# Let's do some calculus! (35)

Calculus Level 5

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 $$f(\cdot)$$.