Compute the derivative of f(x) = (-3x^3 + 7x +11)(2x^2 -4) using product rule

To find the derivative of the function \( f(x) = (-3x^3 + 7x + 11)(2x^2 - 4) \) using the product rule, we proceed as follows:

### Product Rule
The product rule states that if \( f(x) = g(x) \cdot h(x) \), then the derivative is:
\[
f'(x) = g'(x)h(x) + g(x)h'(x)
\]
Here, \( g(x) = -3x^3 + 7x + 11 \) and \( h(x) = 2x^2 - 4 \).

### Step 1: Differentiate \( g(x) \) and \( h(x) \)
- \( g(x) = -3x^3 + 7x + 11 \)
  \[
  g'(x) = -9x^2 + 7
  \]
- \( h(x) = 2x^2 - 4 \)
  \[
  h'(x) = 4x
  \]

### Step 2: Apply the product rule
Using the product rule:
\[
f'(x) = g'(x)h(x) + g(x)h'(x)
\]
Substitute the values of \( g(x) \), \( g'(x) \), \( h(x) \), and \( h'(x) \):
\[
f'(x) = (-9x^2 + 7)(2x^2 - 4) + (-3x^3 + 7x + 11)(4x)
\]

### Step 3: Expand both terms
1. Expand the first term: \( (-9x^2 + 7)(2x^2 - 4) \)
\[
(-9x^2)(2x^2) + (-9x^2)(-4) + 7(2x^2) + 7(-4) = -18x^4 + 36x^2 + 14x^2 - 28 = -18x^4 + 50x^2 - 28
\]

2. Expand the second term: \( (-3x^3 + 7x + 11)(4x) \)
\[
(-3x^3)(4x) + (7x)(4x) + (11)(4x) = -12x^4 + 28x^2 + 44x
\]

### Step 4: Combine all terms
Now combine the two results:
\[
f'(x) = (-18x^4 + 50x^2 - 28) + (-12x^4 + 28x^2 + 44x)
\]
\[
f'(x) = -18x^4 - 12x^4 + 50x^2 + 28x^2 + 44x - 28
\]
\[
f'(x) = -30x^4 + 78x^2 + 44x - 28
\]

Thus, the derivative of the function is:
\[
f'(x) = -30x^4 + 78x^2 + 44x - 28
\]

Would you like further details or explanations on any step?

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Here are 5 related questions:
1. How does the chain rule differ from the product rule?
2. What is the derivative of a product of three functions?
3. How would the derivative change if the function had a quotient?
4. Can you apply the product rule to trigonometric functions?
5. How do you find second derivatives using the product rule?

**Tip:** Always simplify the expressions before differentiating to make the process easier!

Compute the derivative of f(x) = (-3x^3 + 7x + 11)(2x^2 -4) using product rule

Learn how to compute the derivative of f(x) = (-3x^3 + 7x + 11)(2x^2 - 4) using the product rule. This calculus problem includes step-by-step solutions involving the product rule, power rule, and polynomial differentiation, making it suitable for advanced high school or college-level students.

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