https://file.aiquickdraw.com/mathbot/file/1729451691440bmj9b3ou.jpg

To find the derivative of the function:

\[
f(x) = \csc^3(9x - 5)
\]

we'll apply the chain rule and product rule, as this function involves a composition of functions.

### Steps:
1. **Outer function**: The outer function is \(\csc^3(u)\), where \(u = 9x - 5\).
   - The derivative of \(\csc^3(u)\) with respect to \(u\) is found using the chain rule:
   \[
   \frac{d}{du}[\csc^3(u)] = 3\csc^2(u) \cdot (-\csc(u) \cot(u)) = -3\csc^3(u)\cot(u)
   \]
   
2. **Inner function**: The inner function is \(u = 9x - 5\), and its derivative is:
   \[
   \frac{d}{dx}(9x - 5) = 9
   \]

### Putting it together:

Using the chain rule:

\[
f'(x) = \frac{d}{dx} \left[\csc^3(9x - 5)\right]
= -3\csc^3(9x - 5) \cdot \cot(9x - 5) \cdot 9
\]

Simplifying:

\[
f'(x) = -27 \csc^3(9x - 5) \cot(9x - 5)
\]

This is the derivative of \( f(x) \).

---

Do you have any questions or want further details on any part of the process? Here are five follow-up questions:

1. What is the significance of the chain rule in this differentiation?
2. How would you approach the derivative if the function had a different power, say \(\csc^2(9x - 5)\)?
3. What are the key properties of the \(\csc(x)\) and \(\cot(x)\) functions that make their derivatives important in trigonometry?
4. Can you find the second derivative of this function?
5. How does the cotangent function behave for certain values of \(x\) in this context?

**Tip**: When differentiating trigonometric functions raised to powers, always remember to apply both the power rule and the derivative of the trigonometric function itself!

Find the derivative of f(x) = csc^3(9x - 5)

Learn how to find the derivative of the function f(x) = csc^3(9x - 5) using the chain rule and power rule. This step-by-step solution demonstrates differentiation techniques for trigonometric functions. Suitable for undergraduate students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation