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    Question

    The derivative of y = eˣ · cos(x) is 

    A eˣ (cos(x) – sin(x)) Correct Answer Incorrect Answer
    B eˣ (cos(x) + sin(x)) Correct Answer Incorrect Answer
    C eˣ (sin(x) – cos(x)) Correct Answer Incorrect Answer
    D eˣ • sin(x) Correct Answer Incorrect Answer

    Solution

    We have a function
    y = eˣ · cos(x)
    which is a product of two functions:

    • First function: u = eˣ
    • Second function: v = cos(x)
    Step-by-step differentiation using the Product Rule: The product rule states: If y = u · v, then
    dy/dx = (du/dx)·v + u·(dv/dx) Now, differentiate each part: 1.     u = eˣ ⇒ du/dx = eˣ 2.     v = cos(x) ⇒ dv/dx = –sin(x) Now apply the product rule: dy/dx = (eˣ)·cos(x) + (eˣ)·(–sin(x))
        = eˣ·cos(x) – ex·sin(x) Final Answer is eˣ (cos(x) – sin(x))

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