Question

Consider the function f(x) = -18ln(71x). Compute f'(x) and find the exact value of f'(8).

Answer 1

The derivative of f(x) = -18ln(71x) is f'(x) = -18/x. The exact value of the derivative at x = 8 is f'(8) = -2.25.

Step-by-step explanation:

The student has asked to compute the derivative of the function f(x) = -18ln(71x) and to find the exact value of the derivative at x = 8.

Step-by-Step Explanation

1. Begin by using the chain rule to differentiate the function. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.
2. For the logarithmic function ln(g(x)), where g(x) is a function of x, the derivative is: f'(x) = g'(x)/g(x).
3. Applying the chain rule to f(x), we get: f'(x) = -18 * (1/(71x)) * 71; the 71 from the derivative of 71x cancels out with the one in the denominator.
4. Therefore, f'(x) simplifies to f'(x) = -18/x.
5. To find the exact value of f'(8), simply substitute x with 8: f'(8) = -18/8 = -2.25.

Exact Value of f'(8)

The exact value of f'(8) is -2.25.