WEBVTT

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Here is the problem: derivative of e to the power x sin(x).

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Find the derivative of e to the power x sin(x) with respect to x.

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Product rule: (uv) prime equals u prime v plus uv prime.

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d by d x[sin x] equals cos x.

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Exponential rule: d by d x[eᵘ] equals eᵘ times u prime.

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Combine and simplify.

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So the answer is f prime of x equals e to the power x times (sin(x) plus cos(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.