Final answer:
Neither logarithmic differentiation nor any of the provided options are required to find the derivative of the function f(x) = (x + 7)(3x - 8.5). The correct derivative, obtained using the product rule, is 6x + 12.5.
Step-by-step explanation:
To differentiate the function f(x) = (x + 7) (3x – 8.5) using logarithmic differentiation, we would typically take the natural logarithm of both sides of the equation. However, since the function is given as a product of two terms, we do not need logarithmic differentiation in this case. We can simply apply the product rule.
The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Applying the product rule, we have:
f'(x) = (1)(3x - 8.5) + (x + 7)(3) = 3x - 8.5 + 3x + 21
Combining like terms, we get:
f'(x) = 6x + 12.5
Therefore, none of the given options (a), (b), (c), or (d) are correct for the derivative of the function f(x).