Math Problem Statement

Solve

Solution

To solve the problem, we need to find the derivative of the given function y=(2x2+6x)(3x4)y = (2x^2 + 6x)(3x - 4) using the Product Rule. The Product Rule states that if y=uvy = uv, where uu and vv are functions of xx, then the derivative yy' is given by:

y=uv+uvy' = u'v + uv'

Step 1: Identify uu and vv

From the problem:

  • u=2x2+6xu = 2x^2 + 6x
  • v=3x4v = 3x - 4

Step 2: Find the derivatives uu' and vv'

  • u=ddx(2x2+6x)u' = \frac{d}{dx}(2x^2 + 6x) u=4x+6u' = 4x + 6
  • v=ddx(3x4)v' = \frac{d}{dx}(3x - 4) v=3v' = 3

Step 3: Apply the Product Rule

y=uv+uvy' = u'v + uv' Substitute uu, uu', vv, and vv' into the equation: y=(4x+6)(3x4)+(2x2+6x)(3)y' = (4x + 6)(3x - 4) + (2x^2 + 6x)(3)

Step 4: Simplify each term

  • First term: (4x+6)(3x4)(4x + 6)(3x - 4) (4x+6)(3x4)=12x216x+18x24=12x2+2x24(4x + 6)(3x - 4) = 12x^2 - 16x + 18x - 24 = 12x^2 + 2x - 24

  • Second term: (2x2+6x)(3)(2x^2 + 6x)(3) (2x2+6x)(3)=6x2+18x(2x^2 + 6x)(3) = 6x^2 + 18x

Step 5: Combine like terms

y=(12x2+2x24)+(6x2+18x)=18x2+20x24y' = (12x^2 + 2x - 24) + (6x^2 + 18x) = 18x^2 + 20x - 24

Final Answer

The derivative yy' is 18x2+20x2418x^2 + 20x - 24.

Looking at the options provided, this corresponds to the following choice:

  • Option C\boxed{\text{Option C}}

Would you like further details on this solution or have any questions?

Here are some related questions you might find helpful:

  1. What is the general form of the Product Rule, and when is it applicable?
  2. How do you find the derivative of a polynomial function?
  3. What are some common mistakes to avoid when using the Product Rule?
  4. How does the Chain Rule differ from the Product Rule in calculus?
  5. Can the Product Rule be extended to three or more functions?
  6. How do you interpret the result of a derivative in a real-world context?
  7. What is the significance of combining like terms after finding derivatives?
  8. How does the choice of uu and vv affect the calculation in the Product Rule?

Tip: Always double-check your algebraic simplifications when applying the Product Rule to avoid errors in the final derivative.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Product Rule

Formulas

Product Rule for derivatives

Theorems

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Suitable Grade Level

Advanced High School / College