To find the derivative of the function $f(x) = (x^2 - 4x + 2)(5x^3 - x^2 + 4)$, we can apply the product rule. The product rule states that if $f(x) = g(x) \cdot h(x)$, then:

$f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$

Here, we have:

• $g(x) = x^2 - 4x + 2$
• $h(x) = 5x^3 - x^2 + 4$

Step 1: Differentiate $g(x)$ and $h(x)$

1. Differentiate $g(x) = x^2 - 4x + 2$:

$g'(x) = 2x - 4$

2. Differentiate $h(x) = 5x^3 - x^2 + 4$:

$h'(x) = 15x^2 - 2x$

Step 2: Apply the product rule

Now, using the product rule:

$f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$

Substitute the values of $g(x)$, $g'(x)$, $h(x)$, and $h'(x)$:

$f'(x) = (2x - 4)(5x^3 - x^2 + 4) + (x^2 - 4x + 2)(15x^2 - 2x)$

Step 3: Simplify the expression

Now, expand both terms:

1. Expand $(2x - 4)(5x^3 - x^2 + 4)$:

$(2x - 4)(5x^3 - x^2 + 4) = 2x(5x^3 - x^2 + 4) - 4(5x^3 - x^2 + 4)$ $= 10x^4 - 2x^3 + 8x - 20x^3 + 4x^2 - 16$ $= 10x^4 - 22x^3 + 4x^2 + 8x - 16$

2. Expand $(x^2 - 4x + 2)(15x^2 - 2x)$:

$(x^2 - 4x + 2)(15x^2 - 2x) = x^2(15x^2 - 2x) - 4x(15x^2 - 2x) + 2(15x^2 - 2x)$ $= 15x^4 - 2x^3 - 60x^3 + 8x^2 + 30x^2 - 4x$ $= 15x^4 - 62x^3 + 38x^2 - 4x$

Step 4: Combine the results

Now combine the expanded terms from both parts:

$f'(x) = (10x^4 - 22x^3 + 4x^2 + 8x - 16) + (15x^4 - 62x^3 + 38x^2 - 4x)$ $f'(x) = (10x^4 + 15x^4) + (-22x^3 - 62x^3) + (4x^2 + 38x^2) + (8x - 4x) - 16$ $f'(x) = 25x^4 - 84x^3 + 42x^2 + 4x - 16$

Final Answer:

The derivative of $f(x)$ is:

$f'(x) = 25x^4 - 84x^3 + 42x^2 + 4x - 16$

Would you like a more detailed explanation on any part, or do you have other questions?

Here are some related questions for further practice:

1. How would you apply the product rule to functions with more than two factors?
2. What is the chain rule, and how does it differ from the product rule?
3. How do you simplify a complex polynomial expression after differentiation?
4. What is the derivative of a quotient of two functions?
5. How can you find higher-order derivatives of polynomial functions?

Tip: When applying the product rule, always remember to differentiate each function separately and then apply the rule step by step!