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The given problem asks to find the derivative of \( y \) with respect to \( t \), where:

\[
y = \ln(t^6)
\]

### Solution:

First, we can simplify \( \ln(t^6) \) using the logarithmic property:

\[
\ln(a^b) = b \cdot \ln(a)
\]

Thus:

\[
y = 6 \ln(t)
\]

Now, differentiate \( y = 6 \ln(t) \) with respect to \( t \):

\[
\frac{dy}{dt} = 6 \cdot \frac{d}{dt}[\ln(t)]
\]

The derivative of \( \ln(t) \) is \( \frac{1}{t} \), so:

\[
\frac{dy}{dt} = 6 \cdot \frac{1}{t} = \frac{6}{t}
\]

### Final Answer:
\[
\frac{dy}{dt} = \frac{6}{t}
\]

Let me know if you would like more details or have any questions.

Here are five related questions to explore further:
1. How would the derivative change if \( y = \ln(t^n) \) for any \( n \)?
2. What is the second derivative of \( y = \ln(t^6) \)?
3. How does the chain rule apply to logarithmic functions?
4. What happens to the derivative if \( t \) is replaced by another function, like \( t^2 + 1 \)?
5. How can the product rule be used in problems involving logarithms?

**Tip:** Always remember to apply the logarithmic properties to simplify expressions before differentiating, as it can make the process much easier.

Find the derivative of y with respect to t. y = ln(t^6)

Learn how to find the derivative of y = ln(t^6) with respect to t. This problem involves logarithmic differentiation and applies key rules such as the derivative of ln(x) and the power rule. Suitable for advanced high school students in Grades 11-12.

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