In-depth Notes on Product Rule in Derivatives

Introduction to Product Rule

  • The product rule is used to find the derivative of a function that is expressed as a product of two factors.
  • Example function: ( f(x) = (3x + 1)(2x + 3) )

Product Rule Formula

  • If you have ( f ) and ( g ) as two functions, the derivative is calculated as:

    f′=g⋅f′+f⋅g′f' = g \cdot f' + f \cdot g'

  • This formula can be rearranged without affecting the result due to the commutative nature of multiplication and addition.

Step-by-Step Application of Product Rule

  1. Identify the Functions: Assign names to the functions:

    • Let ( f = 3x + 1 ) and ( g = 2x + 3 ).

  2. Find the Derivatives:

    • Derivative of ( f ): ( f' = \frac{d}{dx}(3x + 1) = 3 )
    • Derivative of ( g ): ( g' = \frac{d}{dx}(2x + 3) = 2 )

  3. Apply the Product Rule:

    • Substitute into the formula:

      f′=g⋅f′+f⋅g′  ⟹  (2x+3)⋅3+(3x+1)⋅2f' = g \cdot f' + f \cdot g' \implies (2x + 3) \cdot 3 + (3x + 1) \cdot 2

  4. Distribute and Simplify:

    • First term: ( 6x + 9 )
    • Second term: ( 6x + 2 )
    • Combine like terms: ( 12x + 11 )

Example of Derivative Calculation

  • From the example ( f(x) = (3x + 1)(2x + 3) ), the derivative results in:

    f′(x)=12x+11f'(x) = 12x + 11

Another Example

  • New function: ( f(x) = (4x^2 - 3x + 4)(2x + 3) )
Steps to Find the Derivative:

  1. Function Assignment:

    • Let ( f = 4x^2 - 3x + 4 ) and ( g = 2x + 3 ).

  2. Find Derivatives:

    • ( f' = \frac{d}{dx}(4x^2 - 3x + 4) = 8x - 3 )
    • ( g' = \frac{d}{dx}(2x + 3) = 2 )

  3. Substituting into Formula:

    f′=g⋅f′+f⋅g′  ⟹  (2x+3)(8x−3)+(4x2−3x+4)(2)f' = g \cdot f' + f \cdot g' \implies (2x + 3)(8x - 3) + (4x^2 - 3x + 4)(2)

  4. Distributing:

    • Distribute for both terms
      • First term: ( 16x^2 - 6x + 24x - 9 ) (Using the FOIL method)
      • Second term: ( 8x^2 - 6x + 8 )

  5. Combine Like Terms:

    • Final expression combining both terms:

      (16x2+8x2)+(−6x+24x−6x)+(−9+8)(16x^2 + 8x^2) + (-6x + 24x - 6x) + (-9 + 8)

    • Simplified form: ( 24x^2 + 12x - 1 )

Final Result

  • The derivative of ( f(x) = (4x^2 - 3x + 4)(2x + 3) ) is:

    f′(x)=24x2+12x−1f'(x) = 24x^2 + 12x - 1

Conclusion

  • Mastering the product rule requires practice. Make sure to regularly apply these steps to different functions to solidify understanding!

  • Reminder: Practice makes perfect!