Differentiate the function.
g(t) = ln(6 + t8)

To differentiate the function \( g(t) = \ln(6 + t^8) \), we will use the chain rule.

### Steps:
1. **Differentiate the outer function:** The derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \). So, for \( g(t) = \ln(6 + t^8) \), treat \( 6 + t^8 \) as the inner function.
   \[
   \frac{d}{dt} \left( \ln(6 + t^8) \right) = \frac{1}{6 + t^8} \cdot \frac{d}{dt}(6 + t^8)
   \]

2. **Differentiate the inner function:** The derivative of \( 6 + t^8 \) with respect to \( t \) is \( 8t^7 \) (since \( 6 \) is constant, its derivative is 0).
   \[
   \frac{d}{dt}(6 + t^8) = 8t^7
   \]

3. **Combine the results:** Multiply the derivatives of the outer and inner functions.
   \[
   g'(t) = \frac{1}{6 + t^8} \cdot 8t^7 = \frac{8t^7}{6 + t^8}
   \]

Thus, the derivative of \( g(t) = \ln(6 + t^8) \) is:
\[
g'(t) = \frac{8t^7}{6 + t^8}
\]

Would you like further details or have any questions?

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Here are 5 related questions to explore:
1. What is the derivative of \( g(t) = \ln(t^8 + t^2) \)?
2. How do you apply the chain rule in general cases of logarithmic functions?
3. What happens to the derivative as \( t \to 0 \) or \( t \to \infty \) in the function \( g(t) \)?
4. How would you find the second derivative of \( g(t) = \ln(6 + t^8) \)?
5. What is the derivative of \( g(t) = \ln(a + t^n) \) where \( a \) and \( n \) are constants?

**Tip:** When differentiating composite functions like \( \ln(f(t)) \), always apply the chain rule by differentiating the outer and inner functions separately.

Differentiate the function. g(t) = ln(6 + t^8)

Learn how to differentiate the logarithmic function g(t) = ln(6 + t^8) using the chain rule. This calculus problem demonstrates key differentiation techniques and formulas suitable for Grades 11-12 students. Includes step-by-step explanations and tips for applying the chain rule effectively.

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