Episode 17: The Product Rule and the Quotient Rule

Differentiation Toolbox Review

• Basic Building Block Rules:
  • Definition: ( f' = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} )
  • Derivative of a Constant: 0
  • Power Rule: Derivative of ( x^n ) is ( nx^{n-1} )
  • Constant Multiple Rule: ( (c \cdot f)' = c \cdot f' ) for constant ( c )
  • Sum and Difference Rules: ( (f \pm g)' = f' \pm g' )
  • Chain Rule: Derivative of ( f(g(x)) ) is ( f'(g(x)) \cdot g'(x) )_

New Combination Rules

• Product Rule:

  • Used for differentiating the product of two functions ( f(x) \cdot g(x) )
  • Formula: ( (fg)' = f \cdot g' + g \cdot f' )
  • Proven using the definition of derivative and limits.
  • Example: ( f(x) = -3x \cdot \sqrt{4-x^2} )
    • ( f''(x) ) calculation involves using the product and chain rules.

• Quotient Rule:

  • Used for differentiating the quotient ( \frac{f(x)}{g(x)} )
  • Formula: ( \left(\frac{f}{g}\right)' = \frac{g \cdot f' - f \cdot g'}{g^2} )
  • Example: ( y = \frac{2x-3}{4x+1} )
    • Simplification involves derivatives and algebraic manipulation.

Examples and Applications

• Product Rule Examples:

  • ( f(x) = (x^2 + 3x - 1)(2x^2 - x - 5) )
    • Use product rule to find ( f'(x) )
    • Result: ( f' = 8x^3 + 15x^2 - 20x - 14 )
  • ( f(t) = (3t + 1) \sqrt{t^3 - 5t + 9} )
    • Simplify using product rule: Result involves chain rule for square root function.

• Quotient Rule Examples:

  • ( f(x) = \frac{2x-1}{5x^2 + 7} )
    • Derivatives involve chain and product rules.
  • ( f(x) = \frac{x \sqrt{x^2 + 1}}{2x + 9} )
    • Involves product rule within a quotient rule and simplification.

Observations on Function Behavior

• Points of Inflection:
  • Example of identifying points and intervals of concavity changes.
  • Important to understand the behavior of derived functions.

Conclusion

• With product and quotient rules, differentiation of complex functions is manageable.
• Next steps include studying behavior of functions using these derivative tools.