Question

F(x)=(3x²+2)(x³-4x+6)
Using product rule

Answer 1

To differentiate the function \( F(x) = (3x^2 + 2)(x^3 - 4x + 6) \) using the product rule, we start with the formula for the product rule:

\[
(uv)' = u'v + uv'
\]

Where \( u = 3x^2 + 2 \) and \( v = x^3 - 4x + 6 \).

### Step 1: Differentiate \( u \) and \( v \)

1. Differentiate \( u \):
\[
u' = \frac{d}{dx}(3x^2 + 2) = 6x
\]

2. Differentiate \( v \):
\[
v' = \frac{d}{dx}(x^3 - 4x + 6) = 3x^2 - 4
\]

### Step 2: Apply the product rule

Now, substitute \( u, u', v, \) and \( v' \) into the product rule formula:

\[
F'(x) = u'v + uv'
\]

\[
F'(x) = (6x)(x^3 - 4x + 6) + (3x^2 + 2)(3x^2 - 4)
\]

### Step 3: Simplify each term

1. Expand the first term \( (6x)(x^3 - 4x + 6) \):

\[
6x(x^3) - 6x(4x) + 6x(6) = 6x^4 - 24x^2 + 36x
\]

2. Now expand the second term \( (3x^2 + 2)(3x^2 - 4) \):

\[
3x^2(3x^2) + 3x^2(-4) + 2(3x^2) + 2(-4) = 9x^4 - 12x^2 + 6x^2 - 8
\]

Which simplifies to:
\[
9x^4 - 6x^2 - 8
\]

### Step 4: Combine all terms

Now combine all the terms we have:

\[
F'(x) = (6x^4 - 24x^2 + 36x) + (9x^4 - 6x^2 - 8)
\]

Combine like terms:

1. \( 6x^4 + 9x^4 = 15x^4 \)
2. \( -24x^2 - 6x^2 = -30x^2 \)
3. \( +36x \)
4. Constant term: \( -8 \)

Thus, the final derivative is:

\[
F'(x) = 15x^4 - 30x^2 + 36x - 8
\]

So the derivative \( F'(x) \) is:

\[
\boxed{15x^4 - 30x^2 + 36x - 8}
\]