To differentiate \( Y = \log_5 x \) with respect to \( x \) using the chain rule, we first need to express the logarithm in terms of a more familiar base, such as the natural logarithm (ln).
We can use the change of base formula for logarithms:
\[ Y = \log_5 x = \frac{\ln x}{\ln 5} \]
Now, we can differentiate \( Y \) with respect to \( x \). Since \( \ln 5 \) is a constant, we can factor it out of the derivative:
\[ \frac{dY}{dx} = \frac{1}{\ln 5} \cdot \frac{d}{dx}(\ln x) \]
The derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \). Therefore, we have:
\[ \frac{dY}{dx} = \frac{1}{\ln 5} \cdot \frac{1}{x} \]
Combining these results, the derivative of \( Y = \log_5 x \) with respect to \( x \) is:
\[ \frac{dY}{dx} = \frac{1}{x \ln 5} \]