To find the second derivative of the function [tex]\( f(x) = \log_6(x) \)[/tex], let's go through the process step by step.
### Step 1: Rewrite the Logarithmic Function
The function [tex]\( f(x) = \log_6(x) \)[/tex] can be rewritten in terms of the natural logarithm as:
[tex]\[
f(x) = \frac{\ln(x)}{\ln(6)}
\][/tex]
This is because [tex]\(\log_b(x) = \frac{\ln(x)}{\ln(b)}\)[/tex], where [tex]\(\ln\)[/tex] denotes the natural logarithm.
### Step 2: Find the First Derivative [tex]\( f'(x) \)[/tex]
We start by differentiating [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = \frac{\ln(x)}{\ln(6)}
\][/tex]
To differentiate this, we use the fact that the derivative of [tex]\(\ln(x)\)[/tex] with respect to [tex]\(x\)[/tex] is [tex]\(\frac{1}{x}\)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx} \left( \frac{\ln(x)}{\ln(6)} \right) = \frac{1}{\ln(6)} \cdot \frac{d}{dx} \left( \ln(x) \right) = \frac{1}{\ln(6)} \cdot \frac{1}{x}
\][/tex]
Thus, the first derivative is:
[tex]\[
f'(x) = \frac{1}{x \ln(6)}
\][/tex]
### Step 3: Find the Second Derivative [tex]\( f''(x) \)[/tex]
Next, we need to differentiate [tex]\( f'(x) \)[/tex] to get the second derivative:
[tex]\[
f'(x) = \frac{1}{x \ln(6)}
\][/tex]
To differentiate this, we use the quotient rule or recognize it as a power of [tex]\(x\)[/tex]:
[tex]\[
f'(x) = \frac{1}{x \ln(6)} \Rightarrow f'(x) = \frac{1}{\ln(6)} \cdot x^{-1}
\][/tex]
Now, applying the power rule to differentiate [tex]\(x^{-1}\)[/tex]:
[tex]\[
f''(x) = \frac{1}{\ln(6)} \cdot \frac{d}{dx} (x^{-1})
\][/tex]
[tex]\[
f''(x) = \frac{1}{\ln(6)} \cdot (-1) x^{-2}
\][/tex]
[tex]\[
f''(x) = -\frac{1}{\ln(6)} \cdot \frac{1}{x^2}
\][/tex]
Thus, the second derivative is:
[tex]\[
f''(x) = -\frac{1}{x^2 \ln(6)}
\][/tex]
In conclusion, the second derivative of the function [tex]\( f(x) = \log_6(x) \)[/tex] is:
[tex]\[
f''(x) = -\frac{1}{x^2 \log(6)}
\][/tex]
