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Here is the problem: derivative of ((2x) cubed)/(arctan(x squared plus x plus 1)).

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Find the derivative of ((2x) cubed) / (atan(x squared plus x plus 1)) with respect to x.

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Quotient rule: (u/v) prime equals (u prime v minus uv prime )/v squared.

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Chain rule — d by d x[arctan u] equals u prime /(1 plus u squared).

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Sum rule: differentiate each term separately.

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Sum rule: differentiate each term separately.

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Power rule: d by d x[x to the power n] equals n times x to the power n minus 1.

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Power rule with the chain rule: d by d x[u to the power n] equals n times u to the power n minus 1 times u prime.

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Constant multiple rule.

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Combine and simplify.

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So the answer is f prime of x equals (24x squared plus 8(-(2x to the power 4) minus x cubed) / ((x squared plus x plus 1) squared plus 1) / atan(x squared plus x plus 1)) / atan(x squared plus x plus 1).

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The derivative is built from the outside in: sums split into terms, constants factor out, products and quotients use their rules, and every composite function multiplies by the derivative of its inside (chain rule).

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Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.