Let
f(x) = (1 + 4x2)(x − x2).
Find the derivative by using the Product Rule.
f '(x) =
Find the derivative by multiplying first.
f '(x) =

To find the derivative of the function \( f(x) = (1 + 4x^2)(x - x^2) \), we'll first use the **Product Rule** and then the method of multiplying the terms first before differentiating.

### 1. **Using the Product Rule:**

The Product Rule states that if \( f(x) = g(x) \cdot h(x) \), then the derivative is:
\[
f'(x) = g'(x) h(x) + g(x) h'(x)
\]
For our function:
\[
g(x) = 1 + 4x^2 \quad \text{and} \quad h(x) = x - x^2
\]

#### Step 1: Differentiate \( g(x) \) and \( h(x) \).

- \( g'(x) = \frac{d}{dx}(1 + 4x^2) = 8x \)
- \( h'(x) = \frac{d}{dx}(x - x^2) = 1 - 2x \)

#### Step 2: Apply the Product Rule.

\[
f'(x) = g'(x)h(x) + g(x)h'(x)
\]
Substitute the expressions:
\[
f'(x) = (8x)(x - x^2) + (1 + 4x^2)(1 - 2x)
\]

#### Step 3: Simplify the expression.

First term:
\[
8x(x - x^2) = 8x^2 - 8x^3
\]

Second term:
\[
(1 + 4x^2)(1 - 2x) = 1(1 - 2x) + 4x^2(1 - 2x) = (1 - 2x) + (4x^2 - 8x^3) = 1 - 2x + 4x^2 - 8x^3
\]

Now combine both parts:
\[
f'(x) = (8x^2 - 8x^3) + (1 - 2x + 4x^2 - 8x^3)
\]
\[
f'(x) = 8x^2 - 8x^3 + 1 - 2x + 4x^2 - 8x^3
\]

Combine like terms:
\[
f'(x) = (8x^2 + 4x^2) - 2x + 1 - (8x^3 + 8x^3)
\]
\[
f'(x) = 12x^2 - 2x + 1 - 16x^3
\]

Thus, the derivative is:
\[
f'(x) = -16x^3 + 12x^2 - 2x + 1
\]

### 2. **Multiplying First, Then Differentiating:**

Now, let's multiply the terms first before differentiating.

\[
f(x) = (1 + 4x^2)(x - x^2)
\]

Distribute the terms:
\[
f(x) = 1(x - x^2) + 4x^2(x - x^2)
\]
\[
f(x) = x - x^2 + 4x^2x - 4x^4
\]
\[
f(x) = x - x^2 + 4x^3 - 4x^4
\]

Now differentiate term-by-term:
\[
f'(x) = \frac{d}{dx}(x) - \frac{d}{dx}(x^2) + \frac{d}{dx}(4x^3) - \frac{d}{dx}(4x^4)
\]
\[
f'(x) = 1 - 2x + 12x^2 - 16x^3
\]

Thus, the derivative is the same:
\[
f'(x) = -16x^3 + 12x^2 - 2x + 1
\]

### Final Answer:
\[
f'(x) = -16x^3 + 12x^2 - 2x + 1
\]

Would you like further clarification on any part? Here are five related questions for more practice:

1. What is the Product Rule for derivatives?
2. How do you apply the Chain Rule in differentiation?
3. What is the derivative of \( (3x^4 - 5x^2 + 2x) \)?
4. How do you simplify algebraic expressions involving powers of \( x \)?
5. What is the process for finding second derivatives?

**Tip:** Always double-check your algebra when applying rules like the Product Rule to ensure terms are combined correctly.

Let f(x) = (1 + 4x^2)(x − x^2). Find the derivative by using the Product Rule. f '(x) = Find the derivative by multiplying first. f '(x) =

Find the derivative of f(x) = (1 + 4x^2)(x − x^2) using the Product Rule and by multiplying the terms first. Learn the step-by-step process, including key differentiation rules like the Product Rule and simplifying polynomial functions. Suitable for Grades 11-12.

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