Lecture on the Product Rule

Introduction to the Product Rule

• The product rule is used to find the derivative of the product of two functions.
• Formula: If ( y = f(x) \cdot s(x) ), then the derivative ( y' = f'(x) \cdot s(x) + f(x) \cdot s'(x) ).

Example 1

• Given: ( y = (2x + 1)(3x + 5) ).
  • Treat ( 2x + 1 ) as ( f(x) ) and ( 3x + 5 ) as ( s(x) ).
  • Rewrite equation for clarity: ( y' = (2x + 1)(3x + 5) ).
  • Use different colors for visual aid in calculations.
  • Derivatives:
    • ( 3x + 5 ): Derivative of ( 3x ) is 3, and ( 5 ) is a constant, so derivative is 0.
    • ( 2x + 1 ): Derivative of ( 2x ) is 2, and ( 1 ) is a constant, so derivative is 0.

Solving

• ( y' = (2x + 1) \cdot 3 + (3x + 5) \cdot 2 ).
• Distribute and combine like terms:
  • Calculate: ( 6x + 3 + 6x + 10 = 12x + 13 ).
• Final answer: ( 12x + 13 ).

Example 5

• Given: ( y = (3 - 0.5x^3)(0.4x^2 - 2x) ).
  • Rewrite original function twice for clarity.
  • Derivatives:
    • ( 0.4x^2 ): Multiply 0.4 twice, resulting in ( 0.8x ).
    • ( -2x ): Derivative is ( -2 ).
  • Constant terms (e.g., 3) disappear as they don't have variables.

Solving

• Derivative for the first part combines terms:
  • ( 3 \cdot 0.8x - 2 \cdot 3 = 2.4x - 6 ).
  • ( -0.5x \cdot 0.8 = -0.4x^4 ).

Combining

• Combine like terms:
  • ( -0.4x^4 - 0.6x^4 = -1x^4 ).
  • ( x^3 + 3x^3 = 4x^3 ).
  • ( +2.4x ).
  • Final answer: ( -1x^4 + 4x^3 + 2.4x - 6 ).

Conclusion

• The product rule requires careful application of derivatives and combining like terms.
• Practice is key to mastering derivative calculations and simplifying expressions.