Lecture on Derivative Rules: Product Rule and Quotient Rule

Product Rule Review

• Product Rule Definition: For two functions ( f(x) ) and ( g(x) ), the derivative of their product is given by:
  • ((f \cdot g)' = f'(x) \cdot g(x) + f(x) \cdot g'(x))
  • In each term, differentiate one function and leave the other unchanged.
  • This is a review concept.

Introduction to the Quotient Rule

• Derivation from the Product Rule:
  • Consider ( \frac{f(x)}{g(x)} ).
  • Recognize that this can be expressed as: ( f(x) \cdot g(x)^{-1} ).
• Quotient Rule:
  • Use the product rule and chain rule:
    • ( (f \cdot g^{-1})' = f'(x) \cdot g(x)^{-1} + f(x) \cdot (g(x)^{-1})' )
  • Apply chain rule to ( g(x)^{-1} ):
    • Derivative: (-1 \cdot g(x)^{-2} \cdot g'(x))

Simplifying the Result

• Express the derivative:
  • ( \frac{f'(x)}{g(x)} - \frac{f(x) \cdot g'(x)}{g(x)^2} )
• Combine fractions:
  • To have a common denominator of ( g(x)^2 ):
    • Multiply the first term by ( g(x) ) in the numerator and denominator.
• Final Form:
  • ( \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2} )

Key Observations

• The quotient rule is derived from the product and chain rules.
• Helps in simplifying calculations for derivatives of quotients.
• Compare to product rule:
  • Instead of addition, there is a subtraction.
  • Result is divided by the square of the denominator function.