WEBVTT

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Here is the problem: cos(3x).

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Find the second derivative of cos(3 x) with respect to x.

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Derivative number 1.

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

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Constant multiple rule.

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Simplify the first derivative.

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Derivative number 2.

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Constant multiple rule.

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

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Constant multiple rule.

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Simplify the second derivative.

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So the answer is f prime prime (x) equals -(9cos(3x)).

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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.