WEBVTT

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Here is the problem: d by d x ln(cos(x)).

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Find the derivative of ln(cos(x)) with respect to x.

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Chain rule — d by d x[ln u] equals u prime /u.

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

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Combine and simplify.

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So the answer is f prime of x equals -(sin(x) / cos(x)).

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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.