a. $\frac{d}{dx}(e^x \cos x)$

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Product Rule", "Trigonometric Derivatives", "Exponential Derivatives" ], "current_concepts": [ "Product Rule", "Trigonometric Derivatives", "Exponential Derivatives" ] } ## Identify the functions and rule $$ \begin{aligned} &u(x) = e^x \ &v(x) = \cos x \ &\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \end{aligned} $$ ## Compute individual derivatives $$ \begin{aligned} &u'(x) = \frac{d}{dx}(e^x) = e^x \ &v'(x) = \frac{d}{dx}(\cos x) = -\sin x \end{aligned} $$ ## Apply the product rule and simplify $$ \begin{aligned} \frac{d}{dx}(e^x \cos x) &= e^x \cos x + e^x (-\sin x) \ &= e^x(\cos x - \sin x) \end{aligned} $$ $e^x(\cos x - \sin x)$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Calculus", "Product Rule" ] }

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Explanation:

Identify the functions and rule

Compute individual derivatives

Apply the product rule and simplify

Answer: