https://file.aiquickdraw.com/mathbot/file/1729446918213qicvu7bk.jpg

We are given the function \( y = (5x + 4)^5 (4x + 1)^{-2} \), and we need to find its derivative \( \frac{dy}{dx} \).

### Steps:
We will use the **product rule** combined with the **chain rule** to differentiate this expression. The product rule states that for two functions \( u(x) \) and \( v(x) \), the derivative of their product is:

\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]

Here, let:
- \( u(x) = (5x + 4)^5 \)
- \( v(x) = (4x + 1)^{-2} \)

#### Step 1: Differentiate \( u(x) = (5x + 4)^5 \)
Using the **chain rule**, the derivative of \( u(x) \) is:
\[
u'(x) = 5 \cdot (5x + 4)^4 \cdot \frac{d}{dx}(5x + 4) = 5 \cdot (5x + 4)^4 \cdot 5 = 25 \cdot (5x + 4)^4
\]

#### Step 2: Differentiate \( v(x) = (4x + 1)^{-2} \)
Again, using the **chain rule**, the derivative of \( v(x) \) is:
\[
v'(x) = -2 \cdot (4x + 1)^{-3} \cdot \frac{d}{dx}(4x + 1) = -2 \cdot (4x + 1)^{-3} \cdot 4 = -8 \cdot (4x + 1)^{-3}
\]

#### Step 3: Apply the product rule
Now, apply the product rule:
\[
\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)
\]
Substitute the expressions for \( u'(x) \), \( u(x) \), \( v'(x) \), and \( v(x) \):

\[
\frac{dy}{dx} = 25 \cdot (5x + 4)^4 \cdot (4x + 1)^{-2} + (5x + 4)^5 \cdot (-8 \cdot (4x + 1)^{-3})
\]

This is the derivative of the given function.

Would you like to simplify the expression further, or do you have any questions?

### Related Questions:
1. How would you differentiate the function \( y = (3x + 2)^4 \cdot (x^2 + 1)^{-1} \)?
2. Can you use the product rule if the function has more than two factors?
3. How does the chain rule apply when differentiating powers of linear functions?
4. What is the difference between the product rule and the quotient rule?
5. Can you apply logarithmic differentiation to simplify the differentiation of such functions?

**Tip:** When dealing with products or quotients of functions, always identify which differentiation rules apply (e.g., product, chain, or quotient rule) before proceeding with calculations.

Find the derivative of the function y = (5x + 4)^5 (4x + 1)^{-2}.

Learn how to differentiate the function y = (5x + 4)^5 (4x + 1)^{-2} using the product rule and chain rule. This detailed step-by-step solution covers key calculus concepts and is suitable for students in Grade 11-12 or introductory college 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