Lecture on the Chain Rule

Introduction to the Chain Rule

• Focus on differentiating composite functions: f(g(x))
• Composite function: One function inside another
  • Differentiate the outer function f
  • Multiply by the derivative of the inner function g

Key Concept: Chain Rule Formula

1. Differentiate the outside function (keep inside same)
2. Multiply by the derivative of the inside function

Examples and Applications

Example 1: Polynomial Raised to a Power

• Formula: (u^n)' = n * u^(n-1) * u'
• Example: Derivative of (5x + 3)^4
  • Move exponent (4) to the front
  • Subtract 1 from exponent: (5x + 3)^3
  • Multiply by derivative of inside: 5
  • Final Answer: 20(5x + 3)^3

Example 2: Sine Function

• Example: Derivative of sin(6x)
  • Derivative of sin is cos
  • Multiply by derivative of inside: 6
  • Final Answer: 6cos(6x)

Example 3: Cosine Function

• Example: Derivative of cos(x^2)
  • Derivative of cos is -sin
  • Derivative of inside: 2x
  • Final Answer: -2xsin(x^2)

Example 4: Tangent Function

• Example: Derivative of tan(x^3)
  • Derivative of tan is sec^2
  • Derivative of inside: 3x^2
  • Final Answer: 3x^2sec^2(x^3)

Example 5: Secant Function

• Example: Derivative of sec(4x)
  • Derivative of sec is sec*tan
  • Multiply by derivative of inside: 4
  • Final Answer: 4sec(4x)tan(4x)*

Example 6: Logarithmic Function

• Example: Derivative of (lnx)^7
  • Use power rule (7): 7(lnx)^6
  • Multiply by derivative of lnx: 1/x
  • Final Answer: 7(lnx)^6/x

Example 7: Square Root Function

• Example: Derivative of √(x^3 - 7)
  • Rewrite as (x^3 - 7)^(1/2)
  • Derivative: 3x^2/(2√(x^3 - 7))

Example 8: Quotient with Power

• Example: Derivative of (1/(x^2 + 8)^3)
  • Rewrite: (x^2 + 8)^(-3)
  • Use chain rule and power rule
  • Final Answer: -6x/(x^2 + 8)^4

Advanced Examples

• Nested Trigonometric Functions: Differentiate from outer to inner
• Example: Derivative of sin^5(tan(cos(x^3)))
  • Use chain rule iteratively on nested functions

Combination Rules

• Product and Chain Rule
  • Differentiate using product rule; for complex parts, apply chain rule

Example: Product and Chain Rule

• Example: Derivative of x^3 * (4x + 5)^4
  • Use product rule: Differentiate first, then chain rule on second
  • Simplified Answer: x^2(4x + 5)^3 (28x + 15)*

Quotient Rule with Chain Rule

• Example: [(2x - 3)/(4 + 5x)]^4
  • Use quotient rule within chain rule to find derivative
  • Final Answer: 8(x + 10)(2x - 3)^3/(4 + 5x)^5

Conclusion

• Practice applying the chain rule in various contexts
• Understand the step-by-step differentiation process from outer to inner