WEBVTT

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Here is the problem: d by d x (3x plus 1) to the power 4.

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Find the derivative of (3 x plus 1) to the power 4 with respect to x.

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

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Sum 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 12(3 x plus 1) cubed.

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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.