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Here is the problem: derivative of (sqrt(e to the power x plus 1))(cos(sin(x))).

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Find the derivative of (sqrt(e to the power x plus 1)) (cos(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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Chain rule — d by d x[cos u] equals negative sin u times u prime.

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

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Chain rule — d by d x[the square root of u] equals u prime /(2the square root of u.

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Sum rule: differentiate each term separately.

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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 cos(sin(x)) / 2 minus sqrt(e to the power x plus 1) squared times sin(sin(x)) times cos(x)) / sqrt(e to the power x plus 1).

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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.