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Calculus Level 5

\[ \large f(x)=\frac{1}{1-x+x^2}\]

If \[f_6(x) = \frac ab \sin^c \left( \text{arctan} \left( \frac {\sqrt d}{ex-f} \right) \right) \sin \left( g \cdot \text{arctan} \left( \frac {\sqrt d}{ex-f} \right) \right) \]

where \(a,b,c,d,e,f,g\) are integers independent of \(x\). Evaluate: \(a+b+c+d+e+f+g\).

Details and Assumptions:

  • \(f_n(x)\) denotes \(n^\text{th}\) derivative of \(f(x)\)

  • The greatest common divisor between \(a\) and \(b\) is 1/

  • \(d\) is square free.

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