find the second order derivative of the function g(t)=ln(3t^2+1)

To find the second-order derivative of the function \( g(t) = \ln(3t^2 + 1) \), we'll follow these steps:

### Step 1: First Derivative
We apply the chain rule to find the first derivative.

Given \( g(t) = \ln(3t^2 + 1) \), the derivative of \( \ln(u) \) is \( \frac{1}{u} \), where \( u = 3t^2 + 1 \). We then multiply by the derivative of \( u \) with respect to \( t \), which is \( 6t \) (since \( \frac{d}{dt}(3t^2) = 6t \)).

\[
g'(t) = \frac{1}{3t^2 + 1} \times \frac{d}{dt}(3t^2 + 1) = \frac{1}{3t^2 + 1} \times 6t
\]

So, the first derivative is:

\[
g'(t) = \frac{6t}{3t^2 + 1}
\]

### Step 2: Second Derivative
Now, we need to differentiate \( g'(t) = \frac{6t}{3t^2 + 1} \) using the quotient rule. The quotient rule is:

\[
\frac{d}{dt} \left( \frac{f(t)}{h(t)} \right) = \frac{f'(t)h(t) - f(t)h'(t)}{[h(t)]^2}
\]

In this case:
- \( f(t) = 6t \), so \( f'(t) = 6 \)
- \( h(t) = 3t^2 + 1 \), so \( h'(t) = 6t \)

Now apply the quotient rule:

\[
g''(t) = \frac{(6)(3t^2 + 1) - (6t)(6t)}{(3t^2 + 1)^2}
\]

Simplifying the numerator:

\[
g''(t) = \frac{6(3t^2 + 1) - 36t^2}{(3t^2 + 1)^2}
\]

\[
g''(t) = \frac{18t^2 + 6 - 36t^2}{(3t^2 + 1)^2}
\]

\[
g''(t) = \frac{-18t^2 + 6}{(3t^2 + 1)^2}
\]

### Final Answer:
The second-order derivative of \( g(t) = \ln(3t^2 + 1) \) is:

\[
g''(t) = \frac{-18t^2 + 6}{(3t^2 + 1)^2}
\]

Would you like further details or have any questions?

Here are some related questions:
1. How do you apply the quotient rule to more complex functions?
2. What are the steps for finding higher-order derivatives?
3. How would you find the second derivative of \( g(t) = e^{3t^2 + 1} \)?
4. How do logarithmic functions behave in calculus, especially during differentiation?
5. Can you find the second-order derivative of \( g(t) = \ln(t^2 + 1) \)?

**Tip:** When applying the quotient rule, always carefully differentiate both the numerator and denominator separately before combining them.

Learn how to find the second-order derivative of the function g(t) = ln(3t^2 + 1) using the chain and quotient rules. This detailed solution covers key concepts in calculus, including logarithmic differentiation and the application of the quotient rule. Suitable for advanced high school or undergraduate calculus students.

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