Question

Suppose f(x)=ln(g(x)), where g(x)>0 for all real numbers and is differentiable for all real numbers. If

f(4)=2 and
(4)=−15, find
a) -10
b) -12
c) -14
d) -16

Answer 1

To find the value of g'(4), we need to use the chain rule. Given f(4) = 2 and f'(4) = -15, we can substitute these values into the equation to solve for g'(4). However, none of the options (-10, -12, -14, -16) can be determined based on the given information.

Step-by-step explanation:

To find the value of g'(4), we need to use the chain rule. The chain rule states that if we have a composition of functions, such as f(x) = ln(g(x)), then the derivative of f with respect to x is the derivative of g with respect to x multiplied by the derivative of f with respect to g. In this case, f(x) = ln(g(x)), so f'(x) = g'(x)/g(x).

Given that f(4) = 2 and f'(4) = -15, we can substitute these values into the equation to solve for g'(4):

1. f'(x) = g'(x)/g(x)
2. -15 = g'(4)/g(4)
3. g'(4) = -15 * g(4)

Since we don't have any specific information about g(4), we can't determine the exact value of g'(4). So none of the options (-10, -12, -14, -16) can be determined based on the given information.