# Differentiating unusual

Calculus Level 5

$\large f(x)=\frac{1}{1+x+x^2}$

For the function $$f(x)$$ as described above, then $${\dfrac{d^{6} f(x)}{dx^6}}$$ can be represented as

$\frac ab \sin^c \left( \text{arctan} \left( \frac {\sqrt d}{ex+f} \right) \right) \sin \left( g \cdot \text{arctan} \left( \frac {\sqrt d}{ex+f} \right) \right)$

where $$a,b,c,d,e,f,g$$ are integers independent of $$x$$. for $$\gcd(a,b) = 1$$ and $$d$$ square free.

Evaluate $$a+b+c+d+e+f+g$$.

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