Calculus: Chain Rule, Product Rule, Quotient Rule Study Notes

Chain Rule

• Introduction to the Chain Rule
  • The Chain Rule is used to find the derivative of composite functions.
  • Procedure:
  1. Take the derivative of the outside function.
  2. Multiply it by the derivative of the inside function.
• Example Problem: Derivative Calculation
  • If given a function in parentheses:
  • Step 1: Identify the outside and inside functions.
  • Step 2: Apply the chain rule formula:
    $f(g(x))' = f'(g(x)) imes g'(x)$
  • Example:
  • Derivative of: $12(x^5)$
    • Derivative of outside (12): $12 imes 5 = 60$
    • Multiply by the derivative of the inside (5x^4): $60 imes 2x$
    • Final answer: $120x(x^2 + 6)^4$
    • Important notes: Do not distribute the power beyond one in case of n > 1.

Product Rule and Chain Rule Combined

• Re-introduction to Product and Chain Rules
  • When functions are multiplied together, the Product Rule applies:
  1. Derivative of the first times the second plus the second times the derivative of the first.
  • Example Function: $y = 9x(x + 7)^2$
• Steps to Differentiate:
  • Step 1: Identify parts of the product.
  • Step 2: Apply product rule.
  • Step 3: Apply chain rule where necessary:
  • Derivative of the outside: bring down the exponent 2, leave the inside unchanged, and multiply by the derivative of the inside (which is 1).
• Final Simplification:
  • Identify common factors and factor out:
  • Common factors: $9$, $(x + 7)$.
  • Result: $9(x + 7)(2x + 1)$
  • Further simplification yields: $4(x + 3)$.

Quotient Rule

• Explanation of the Quotient Rule
  • The Quotient Rule is applied when one function is divided by another.
  • Formula:
    $\frac{f}{g} = \frac{g imes f' - f imes g'}{g^2}$
• Example Function:
  • $\frac{5x + 2}{11x}$
• Steps to Differentiate:
  • Apply the quotient rule:
  1. Find the derivative of the numerator (top function) using the chain rule where necessary.
  2. Multiply by the denominator and apply the final simplification.
  • Final result:
  • Identify common terms and reduce: $15x - (5x + 2)$ results in $15x - 5x - 2$.
    • Final answer: $9x - 2$ over the denominator squared.

Homework Instructions

• Students are instructed to complete similar problems as practice for applying chain rule, product rule, and quotient rule.
• Reminder of due dates for related assignments.

Summary

• Key concepts learnt involved applying chain, product, and quotient rules in differentiation, focusing on details like maintaining the expression structure when powers are involved, and proper factorization.
• Engagement with participation asking students if they have any questions before concluding the session.