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.