Question

Suppose f(x) = ln(f(x)), where f(x) > 0 for all real numbers and is differentiable for all real numbers. If f(4) = 2 and f'(4) = -15, find g'(4).

Answer 1

g'(4)=-15/2

To find g'(4), apply the chain rule to differentiate f(x) = ln(f(x)). Plug in the given values to find f'(4) and calculate g'(4).

Step-by-step explanation:

To find g'(4), we need to use the chain rule and the given information about f'(x). Let's start by differentiating f(x) = ln(f(x)).

Using the chain rule, the derivative of ln(f(x)) with respect to x is 1/f(x) * f'(x). Now let's plug in the given values:

f(4) = 2 and f'(4) = -15

So the derivative of f(x) at x=4, denoted as f'(4), is 1/2 * -15 = -15/2.

Therefore, g'(4) is also equal to -15/2.