Review of Derivative Rules in Calculus

Derivative Rules Overview

• Power Rule
• Constant Multiple Rule
• Sum Rule
• Difference Rule
• Product Rule
• Quotient Rule
• Chain Rule

Examples and Application of Derivative Rules

Example 1: Using Power Rule and Constant Multiple Rule

• Function: (-x^5 + 5x^3 + \sqrt[3]{x^2})
• Rewrite (\sqrt[3]{x^2}) as (x^{2/3})
• Apply Power Rule:
  • Derivative of (-x^5) is (-5x^4)
  • Derivative of (5x^3) is (15x^2)
  • Derivative of (x^{2/3}) is (\frac{2}{3}x^{-1/3})
• Final derivative: (-5x^4 + 15x^2 + \frac{2}{3x^{1/3}})

Example 2: Using Product Rule

• If (f(x) = 11x + 2) and (g(x) = -5 + 3x^2)
• Product Rule: (f'(x)g(x) + g'(x)f(x))
• Derivative: (99x^2 + 12x - 55)

Example 3: Using Quotient Rule

• Derivative of a quotient ((f/g)' = (g'f - fg')/g^2)
• Function: ((x^2 - 4)/(2x + 5))
• Derivative: (\frac{2x^2 + 10x + 8}{(2x + 5)^2})

Example 4: Using Chain Rule

• Function: ((3x - 2)^2)
• Chain Rule: Differentiate outer function, multiply by derivative of inner function
• Derivative: (6(3x - 2))

Example 5: Combination of Chain Rule and Product Rule

• Use product rule with functions that require chain rule
• Simplify using common factors
• Example: (2x - 3)^3 \times (3x - 1)^2)
• Final Derivative: (6(2x - 3)^2(3x - 1)(5x - 4))

Example 6: Combination of Quotient Rule and Chain Rule

• Function: (\frac{8x^3}{\sqrt{3x - 2}})
• Factor out common terms to simplify
• Final derivative in simplified form

Application: Finding Equation of Tangent Line

Example: Tangent Line at a Point

• Function: (y = \frac{2x}{(x + 1)^6})
• Find tangent at (x = 1)
• Calculate (y(1) = 1)
• Use derivative to find slope (m = 3)
• Equation of tangent line: (y = 3x - 2)

Conclusion

• Review of main derivative rules from high school calculus
• Importance of understanding and practicing each rule
• Refer to individual topic videos for more detailed explanations