Final Answer:
Presuming that \( g(x) \) was meant to be mentioned in the problem, let's assume the function \( g(x) \) is defined by:
\[ g(x) = \ln(f(x)) \] \( g'(4) = -\frac{1}{10} \), which corresponds to option D
Step-by-step explanation:
To solve this question, we need to clarify the probable mistake in the given equation \( f(x) = \ln(f(x)) \), since it doesn't make sense for \( f(x) \) to be equal to its natural logarithm in general.
We are given \( f(4) = 2 \) and \( f'(4) = -\frac{1}{5} \), and we want to find the derivative of \( g(x) \) at \( x = 4 \). We can find \( g'(x) \) by applying the chain rule for differentiation, which states that if \( g(x) = h(f(x)) \), then
\[ g'(x) = h'(f(x)) \cdot f'(x) \]
In our case, \( h(x) \) would be the natural logarithm function \( \ln(x) \), therefore
\[ h'(x) = \frac{1}{x} \]
Now, we apply the chain rule for the function \( g(x) = \ln(f(x)) \):
\[ g'(x) = h'(f(x)) \cdot f'(x) = \frac{1}{f(x)} \cdot f'(x) \]
We substitute \( x = 4 \):
\[ g'(4) = \frac{1}{f(4)} \cdot f'(4) \]
We were given:
\[ f(4) = 2 \quad \text{and} \quad f'(4) = -\frac{1}{5} \]
Substitute these values in the expression for \( g'(4) \):
\[ g'(4) = \frac{1}{2} \cdot \left(-\frac{1}{5}\right) = -\frac{1}{2} \cdot \frac{1}{5} = -\frac{1}{10} \]
Therefore, \( g'(4) = -\frac{1}{10} \), which corresponds to option D