find the derivative of the function $c(x) = 7e^{3\sin x}$.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Derivative", "Trigonometric Derivatives" ], "new_concepts": [ "Chain Rule Differentiation" ], "current_concepts": [ "Exponential Derivative", "Trigonometric Derivatives", "Chain Rule Differentiation" ] } ## Identify the function and differentiation rules $$ C(x) = 7e^{3\sin x} $$ ## Apply the chain rule $$ C'(x) = 7 \cdot e^{3\sin x} \cdot \frac{d}{dx}(3\sin x) $$ ## Compute the inner derivative and simplify $$ \begin{aligned} \frac{d}{dx}(3\sin x) &= 3\cos x \ C'(x) &= 7e^{3\sin x} \cdot (3\cos x) \ C'(x) &= 21\cos(x)e^{3\sin(x)} \end{aligned} $$ $C'(x) = 21\cos(x)e^{3\sin(x)}$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Calculus", "Chain Rule Differentiation" ] }

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Explanation:

Identify the function and differentiation rules

Apply the chain rule

Compute the inner derivative and simplify

Answer: