WEBVTT

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Here is the problem: derivative of (ln(sin(x))) plus (arctan(2x)).

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Find the derivative of (ln(sin(x))) plus (atan(2 x)) with respect to x.

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Sum rule: differentiate each term separately.

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Chain rule — d by d x[arctan u] equals u prime /(1 plus u squared).

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Constant multiple rule.

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

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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 cos(x) / sin(x) plus 2 / (4x squared 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.