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Let y = uv be the product of the functio...

Let y = uv be the product of the functions `u and v`. Find `y'(2)` if `u(2) = 3, u'(2) = – 4, v(2) = 1, and v'(2) = 2.`

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To find \( y'(2) \) where \( y = uv \), we will use the product rule of differentiation. The product rule states that if \( y = uv \), then: \[ y' = u'v + uv' \] ### Step-by-Step Solution: 1. **Identify the functions and their values:** - Given: - \( u(2) = 3 \) - \( u'(2) = -4 \) - \( v(2) = 1 \) - \( v'(2) = 2 \) 2. **Apply the product rule:** - Using the product rule, we can express \( y'(2) \) as: \[ y'(2) = u(2)v'(2) + v(2)u'(2) \] 3. **Substitute the known values:** - Plugging in the values we have: \[ y'(2) = u(2)v'(2) + v(2)u'(2) = (3)(2) + (1)(-4) \] 4. **Calculate each term:** - Calculate \( 3 \times 2 = 6 \) - Calculate \( 1 \times -4 = -4 \) 5. **Combine the results:** - Now, combine the results: \[ y'(2) = 6 - 4 = 2 \] ### Final Answer: Thus, \( y'(2) = 2 \). ---
To find \( y'(2) \) where \( y = uv \), we will use the product rule of differentiation. The product rule states that if \( y = uv \), then: \[ y' = u'v + uv' \] ### Step-by-Step Solution: ...
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