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F(x)=(3x²+2)(x³-4x+6)

Using product rule

1 answer

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:

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