https://file.aiquickdraw.com/mathbot/file/17299141813935s00l8gx.jpg

To find the derivative of the function 

\[
y = \frac{4x^2 - x}{(3x - 1)^7},
\]

we'll use the **quotient rule** combined with the **chain rule**.

### Step-by-Step Solution

The quotient rule for a function \( y = \frac{u}{v} \) is:

\[
\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]

where:
- \( u = 4x^2 - x \)
- \( v = (3x - 1)^7 \)

We need to compute:
1. \( \frac{du}{dx} \),
2. \( \frac{dv}{dx} \).

#### 1. Differentiate \( u = 4x^2 - x \)
\[
\frac{du}{dx} = 8x - 1
\]

#### 2. Differentiate \( v = (3x - 1)^7 \) using the chain rule
Let \( w = 3x - 1 \). Then \( v = w^7 \), and applying the chain rule, we get:
\[
\frac{dv}{dx} = 7(3x - 1)^6 \cdot \frac{d}{dx}(3x - 1)
\]
Since \( \frac{d}{dx}(3x - 1) = 3 \), we have:
\[
\frac{dv}{dx} = 7(3x - 1)^6 \cdot 3 = 21(3x - 1)^6
\]

#### 3. Apply the Quotient Rule
Now, substitute into the quotient rule formula:
\[
\frac{dy}{dx} = \frac{(3x - 1)^7 \cdot (8x - 1) - (4x^2 - x) \cdot 21(3x - 1)^6}{((3x - 1)^7)^2}
\]

Simplify the denominator:
\[
((3x - 1)^7)^2 = (3x - 1)^{14}
\]

So,
\[
\frac{dy}{dx} = \frac{(3x - 1)^7 (8x - 1) - (4x^2 - x) \cdot 21(3x - 1)^6}{(3x - 1)^{14}}
\]

Now, factor out \( (3x - 1)^6 \) in the numerator:
\[
\frac{dy}{dx} = \frac{(3x - 1)^6 \left[(3x - 1)(8x - 1) - 21(4x^2 - x)\right]}{(3x - 1)^{14}}
\]

Finally, cancel \( (3x - 1)^6 \) from the numerator and denominator:
\[
\frac{dy}{dx} = \frac{(3x - 1) \cdot (8x - 1) - 21(4x^2 - x)}{(3x - 1)^8}
\]

This is the derivative of the given function.

Would you like a more detailed simplification or have any questions?

### Related Questions

1. How is the chain rule applied in differentiation of composite functions?
2. What is the purpose of the quotient rule in differentiation?
3. Can you explain the process of factoring terms in derivatives?
4. How does differentiating powers of functions differ from simple functions?
5. What are other examples of derivatives that combine chain and quotient rules?

### Tip
When applying both quotient and chain rules, break the process into manageable steps to avoid errors, especially in long expressions.

Find the derivative of the function y = (4x^2 - x) / (3x - 1)^7.

Find the derivative of y = (4x^2 - x) / (3x - 1)^7 with a detailed solution applying the quotient rule and chain rule. This calculus problem explores key differentiation concepts suitable for Grade 12 or college Calculus I students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation