WEBVTT

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Here is the problem: differentiate cos(x squared).

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

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

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Power rule: d by d x[x to the power n] equals n times x to the power n minus 1.

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Combine and simplify.

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So the answer is f prime of x equals -(2sin(x squared) times 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.