https://file.aiquickdraw.com/mathbot/file/1728510220412n4tken1c.jpg

The function provided is \( f(t) = \frac{1}{2}(9t^2 + t)^{-3} \). We are tasked with finding the derivative \( f'(t) \).

### Steps to find the derivative:

We will apply the **chain rule** and the **power rule** for differentiation.

1. **Start with the outer function**:
   The outer function is \( (9t^2 + t)^{-3} \), and the derivative of \( x^{-n} \) is \( -n x^{-n-1} \). 
   - So, the derivative of \( (9t^2 + t)^{-3} \) with respect to \( (9t^2 + t) \) is:
     \[
     -3(9t^2 + t)^{-4}
     \]

2. **Multiply by the derivative of the inner function**:
   The inner function is \( 9t^2 + t \), and its derivative with respect to \( t \) is:
   \[
   \frac{d}{dt}(9t^2 + t) = 18t + 1
   \]

3. **Combine the results**:
   Now multiply the outer derivative by the derivative of the inner function:
   \[
   f'(t) = \frac{1}{2} \times -3(9t^2 + t)^{-4} \times (18t + 1)
   \]

4. **Simplify the expression**:
   \[
   f'(t) = -\frac{3}{2}(9t^2 + t)^{-4}(18t + 1)
   \]

Thus, the derivative \( f'(t) \) is:
\[
f'(t) = -\frac{3}{2}(9t^2 + t)^{-4}(18t + 1)
\]

Would you like further clarification or details on any of the steps? 

Here are some related questions:
1. How do you apply the chain rule in other examples?
2. What happens if the exponent in the function was different (e.g., \( (9t^2 + t)^{-2} \))?
3. What are the basic rules for differentiating power functions?
4. How does the product rule differ from the chain rule in differentiation?
5. Can the quotient rule be applied here? Why or why not?

**Tip:** When using the chain rule, always differentiate the outer function first and then multiply by the derivative of the inner function.

Find the derivative of the function: f(t) = (1/2) * (9t^2 + t)^(-3).

Learn how to find the derivative of f(t) = (1/2) * (9t^2 + t)^(-3) using the chain and power rules. This calculus problem includes a step-by-step solution with explanations of differentiation techniques. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation