WEBVTT

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Here is the problem: d by d x((3x plus 1)/(x minus 2)).

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Find the derivative of ((3 x plus 1) / (x minus 2)) 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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Difference rule: differentiate each term separately.

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Sum 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 ((negative 3x minus 1) / (x minus 2) plus 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.