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We are tasked with finding the derivative \( \frac{dy}{dx} \) of the given function:

\[
y = 5 \log_6{x}
\]

### Step 1: Convert the logarithmic base
To make differentiation easier, we can convert the logarithmic base to natural logarithms using the following identity:

\[
\log_b{x} = \frac{\ln{x}}{\ln{b}}
\]

So,

\[
\log_6{x} = \frac{\ln{x}}{\ln{6}}
\]

Thus, the given equation becomes:

\[
y = 5 \cdot \frac{\ln{x}}{\ln{6}}
\]

### Step 2: Differentiate using the chain rule
We can now differentiate the function \( y = \frac{5}{\ln{6}} \cdot \ln{x} \) with respect to \(x\).

The derivative of \( \ln{x} \) is \( \frac{1}{x} \), so we have:

\[
\frac{dy}{dx} = \frac{5}{\ln{6}} \cdot \frac{1}{x}
\]

### Final Answer:
\[
\frac{dy}{dx} = \frac{5}{x \ln{6}}
\]

This is the derivative of \( y = 5 \log_6{x} \).

Would you like a more detailed explanation or have any questions?

---

Here are 5 related questions to expand your understanding:

1. What is the derivative of \( y = \log_b{x} \) for any base \( b \)?
2. How do you apply the chain rule in logarithmic differentiation?
3. How does the change of base formula for logarithms help simplify derivatives?
4. How would you differentiate \( y = \log_6{(2x+1)} \)?
5. Can you derive the second derivative \( \frac{d^2y}{dx^2} \) of the given function?

**Tip:** When differentiating logarithms with bases other than \( e \) (natural logarithms), always use the change of base formula to convert them into terms involving \( \ln{x} \).

Learn how to differentiate y = 5 log_6(x) using the change of base formula and logarithmic differentiation. The solution includes the conversion of the logarithmic base and applying the chain rule, suitable for high school calculus students.

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