Use the product rule to find the derivative of the function.
y equals left parenthesis 2 x squared plus 3 right parenthesis left parenthesis 3 x minus 4 right parenthesis


To find the derivative of the function \( y = (2x^2 + 3)(3x - 4) \) using the **Product Rule**, we proceed as follows:

The Product Rule states:
\[
\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]

Here, let:
- \( f(x) = 2x^2 + 3 \)
- \( g(x) = 3x - 4 \)

### Step 1: Differentiate \( f(x) \) and \( g(x) \)

1. Derivative of \( f(x) = 2x^2 + 3 \):
   \[
   f'(x) = \frac{d}{dx}(2x^2 + 3) = 4x
   \]

2. Derivative of \( g(x) = 3x - 4 \):
   \[
   g'(x) = \frac{d}{dx}(3x - 4) = 3
   \]

### Step 2: Apply the Product Rule

Using the Product Rule:
\[
\frac{d}{dx} [y] = f'(x)g(x) + f(x)g'(x)
\]

Substitute \( f(x), g(x), f'(x), \) and \( g'(x) \) into the equation:
\[
\frac{d}{dx} [y] = (4x)(3x - 4) + (2x^2 + 3)(3)
\]

### Step 3: Simplify the expression

1. Distribute \( 4x \) into \( (3x - 4) \):
   \[
   4x(3x - 4) = 12x^2 - 16x
   \]

2. Distribute \( 3 \) into \( (2x^2 + 3) \):
   \[
   3(2x^2 + 3) = 6x^2 + 9
   \]

### Step 4: Add the results

Now, combine the terms:
\[
\frac{d}{dx} [y] = (12x^2 - 16x) + (6x^2 + 9)
\]
\[
\frac{d}{dx} [y] = 18x^2 - 16x + 9
\]

Thus, the derivative of the function \( y = (2x^2 + 3)(3x - 4) \) is:
\[
\boxed{18x^2 - 16x + 9}
\]

Would you like more details on any part of this process? Here are 5 related questions to extend your understanding:

1. What happens if you apply the quotient rule instead of the product rule for similar problems?
2. How would the result change if the function included higher powers of \( x \)?
3. What is the second derivative of the function \( y = (2x^2 + 3)(3x - 4) \)?
4. How can you factor the derivative \( 18x^2 - 16x + 9 \)?
5. Can you apply the chain rule to this function if additional terms were inside parentheses?

**Tip**: Always check if the terms in your derivative can be simplified further to make interpretation easier.

Use the product rule to find the derivative of the function. y equals left parenthesis 2 x squared plus 3 right parenthesis left parenthesis 3 x minus 4 right parenthesis

Learn how to apply the product rule to find the derivative of the function y = (2x^2 + 3)(3x - 4) using step-by-step instructions. This example covers key concepts in differentiation and the application of the product rule, suitable for students in Grades 11-12 and early college.

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