https://file.aiquickdraw.com/mathbot/file/1726456389046so3bwa7i.jpg

The function to differentiate is:

\[
h(t) = \left( \frac{t^3}{t^6 + 9} \right)^2
\]

To find \( h'(t) \), we will use the chain rule along with the quotient rule. Here are the steps:

1. **Chain Rule**: First, recognize that the outer function is a square, so we will differentiate \( (\text{something})^2 \) first and then multiply by the derivative of the inside.
   
   Using the chain rule: 
   \[
   \frac{d}{dt} \left( u^2 \right) = 2u \cdot \frac{du}{dt}
   \]
   where \( u = \frac{t^3}{t^6 + 9} \).

2. **Quotient Rule**: Now, to differentiate the inside part \( \frac{t^3}{t^6 + 9} \), we use the quotient rule:
   \[
   \frac{d}{dt} \left( \frac{f(t)}{g(t)} \right) = \frac{f'(t)g(t) - f(t)g'(t)}{(g(t))^2}
   \]
   where \( f(t) = t^3 \) and \( g(t) = t^6 + 9 \).

   Let's calculate the necessary derivatives:
   - \( f'(t) = 3t^2 \)
   - \( g'(t) = 6t^5 \)

   Substituting these into the quotient rule:
   \[
   \frac{d}{dt} \left( \frac{t^3}{t^6 + 9} \right) = \frac{3t^2(t^6 + 9) - t^3(6t^5)}{(t^6 + 9)^2}
   \]
   Simplifying the numerator:
   \[
   3t^2(t^6 + 9) - t^3(6t^5) = 3t^8 + 27t^2 - 6t^8 = -3t^8 + 27t^2
   \]
   So the derivative becomes:
   \[
   \frac{-3t^8 + 27t^2}{(t^6 + 9)^2}
   \]

3. **Final derivative**: Now applying the chain rule to get \( h'(t) \):
   \[
   h'(t) = 2 \cdot \left( \frac{t^3}{t^6 + 9} \right) \cdot \frac{-3t^8 + 27t^2}{(t^6 + 9)^2}
   \]
   Simplifying the final expression:
   \[
   h'(t) = \frac{2t^3(-3t^8 + 27t^2)}{(t^6 + 9)^3}
   \]

So, the derivative is:

\[
h'(t) = \frac{2t^3(-3t^8 + 27t^2)}{(t^6 + 9)^3}
\]

Would you like a more detailed explanation of any part of this solution?

---

**Related questions:**
1. How does the chain rule work in this context?
2. Can you explain the quotient rule in more depth?
3. What would change if the power was different, such as \( \left( \frac{t^3}{t^6 + 9} \right)^3 \)?
4. How can the quotient rule be generalized for more complex functions?
5. What are some common mistakes to avoid when applying the quotient rule?

**Tip:** Always simplify the numerator and denominator before differentiating to make the process easier.

Find the derivative of the function h(t) = (t^3 / (t^6 + 9))^2.

Learn how to find the derivative of the function h(t) = (t^3 / (t^6 + 9))^2 by applying the Chain Rule and the Quotient Rule. This detailed solution involves differentiating a composite function and simplifying the result.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation