Find the derivative of the given function.

𝑦=(3𝑥−5)^-1

To find the derivative of the function \( y = (3x - 5)^{-1} \), we can use the **chain rule**.

### Step-by-step solution:

1. **Rewrite the function:**
   \[
   y = (3x - 5)^{-1}
   \]
   This is equivalent to:
   \[
   y = \frac{1}{3x - 5}
   \]

2. **Apply the chain rule:**
   The derivative of \( (3x - 5)^{-1} \) with respect to \( x \) involves the chain rule, since \( 3x - 5 \) is an inner function.

   - The outer function is \( u^{-1} \), where \( u = 3x - 5 \).
   - The derivative of \( u^{-1} \) with respect to \( u \) is \( -u^{-2} \).

3. **Differentiate the inner function:**
   The derivative of \( 3x - 5 \) with respect to \( x \) is \( 3 \).

4. **Combine the results using the chain rule:**
   \[
   \frac{dy}{dx} = -\frac{1}{(3x - 5)^2} \times 3
   \]

5. **Simplify the derivative:**
   \[
   \frac{dy}{dx} = -\frac{3}{(3x - 5)^2}
   \]

### Final Answer:
\[
\frac{dy}{dx} = -\frac{3}{(3x - 5)^2}
\]

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What is the derivative of \( y = (5x + 7)^{-1} \)?
2. How would you differentiate \( y = (2x^2 - 4x + 1)^{-1} \)?
3. What is the second derivative of \( y = (3x - 5)^{-1} \)?
4. How does the chain rule apply when differentiating composite functions?
5. How would you find the derivative of \( y = (3x - 5)^{-3} \)?

**Tip:** When differentiating powers of functions, always remember to apply the chain rule if there's an inner function involved.

Find the derivative of the given function: y = (3x - 5)^-1

Learn how to differentiate the function y = (3x - 5)^-1 using the chain rule. This step-by-step solution covers the chain rule, power rule, and provides related practice problems. Ideal for students in Grades 11-12 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