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Here is the problem: derivative of (sqrt(3x minus 2))(cos(3x minus 2)).

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Find the derivative of (sqrt(3 x minus 2)) (cos(3 x minus 2)) with respect to x.

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Product rule: (uv) prime equals u prime v plus uv prime.

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Chain rule — d by d x[cos u] equals negative sin u times u prime.

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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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Difference rule: differentiate each term separately.

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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 (cos(3 x minus 2) times 3 over 2 minus 3sqrt(3 x minus 2) squared times sin(3 x minus 2)) / sqrt(3 x minus 2).

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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.