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Here is the problem: derivative of (e to the power sin(x)))/(sqrt(x squared plus x plus 1)).

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Find the derivative of (e to the power sin(x))) / (sqrt(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[the square root of u] equals u prime /(2the square root of u.

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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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Exponential rule: d by d x[eᵘ] equals eᵘ times u prime.

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d by d x[sin x] equals cos x.

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Combine and simplify.

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So the answer is f prime of x equals ((negative 2x minus 1) / 2 plus sqrt(x squared plus x plus 1) squared times cos(x)) times e to the power sin(x) / sqrt(x squared plus x plus 1) cubed.

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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.