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Here is the problem: derivative of (ln(3x minus 2))/(cos(e to the power x plus 1)).

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Find the derivative of (ln(3 x minus 2)) / (cos(e to the power 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[cos u] equals negative sin u times u prime.

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Sum rule: differentiate each term separately.

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

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Chain rule — d by d x[ln u] equals u prime /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 (3 / (3 x minus 2) plus ln(3 x minus 2) times sin(e to the power x plus 1) times e to the power x / cos(e to the power x plus 1)) / cos(e to the power 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.