https://file.aiquickdraw.com/mathbot/file/1726249251034mbf41any.jpg

To find the derivative of the given function:

\[
f(x) = x(4x - 9)^3
\]

We will apply the **product rule** and the **chain rule**.

### Step 1: Identify the two parts of the product.
The function is the product of two functions:
- \( u(x) = x \)
- \( v(x) = (4x - 9)^3 \)

The product rule for derivatives states that:

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

### Step 2: Differentiate \( u(x) \) and \( v(x) \).

- \( u(x) = x \), so \( u'(x) = 1 \).
  
Now for \( v(x) = (4x - 9)^3 \), we apply the **chain rule**:
- First, differentiate the outer function \( (z)^3 \), where \( z = 4x - 9 \), to get \( 3(z)^2 \).
- Then, differentiate the inner function \( 4x - 9 \), which gives \( 4 \).

Thus,

\[
v'(x) = 3(4x - 9)^2 \cdot 4 = 12(4x - 9)^2
\]

### Step 3: Apply the product rule.
Using the product rule:

\[
f'(x) = u'(x) v(x) + u(x) v'(x)
\]

Substitute the values we calculated:

\[
f'(x) = 1 \cdot (4x - 9)^3 + x \cdot 12(4x - 9)^2
\]

Simplify:

\[
f'(x) = (4x - 9)^3 + 12x(4x - 9)^2
\]

This is the derivative of the function.

---

Would you like a more detailed explanation of any step? Here are some follow-up questions for further practice:

1. How would you apply the product rule to a different function, like \( f(x) = x^2(5x + 3)^4 \)?
2. How does the chain rule work for functions like \( (7x - 2)^5 \)?
3. What happens if you apply the quotient rule instead of the product rule to a similar function?
4. Can you expand \( (4x - 9)^3 \) and find the derivative directly without using the chain rule?
5. How do you find higher-order derivatives for this function?

**Tip**: When dealing with complex products, always break the function into smaller parts and apply the appropriate rules one step at a time.

Find the derivative of the function f(x) = x(4x - 9)^3

This page provides a step-by-step solution to finding the derivative of the function f(x) = x(4x - 9)^3. It covers the application of the product and chain rules, key calculus concepts essential for understanding complex derivatives.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation