WEBVTT

1
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Here is the problem: differentiate tan(2x).

2
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Find the derivative of tan(2 x) with respect to x.

3
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Chain rule — d by d x[tan u] equals sec squared u times u prime.

4
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Constant multiple rule.

5
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Combine and simplify.

6
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So the answer is f prime of x equals 2sec(2x) squared.

7
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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).

8
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Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.