3.5 Higher order derivatives

Higher Order Derivatives

Definition

• Function Definition: For a polynomial function, f(x) = 2x³ − 18x² + 48x − 27.

• First Derivative:

  • Notation: f′(x) or df/dx

  • Calculation: f′(x) = 6x² − 36x + 48.

Higher Order Derivatives

• Second Derivative:

  • Notation: f′′(x) or d²f/dx²

  • Calculation: f′′(x) = 12x − 36.

• Notation Explanation:

  • f′(x) = df/dx: first derivative of f

  • f′′(x) = d²f/dx²: second derivative of f

  • f′′′(x) = d³f/dx³: third derivative of f

  • f(⁴)(x) = d⁴f/dx⁴: fourth derivative of f

  • f(n)(x) = dⁿf/dxⁿ: nth derivative of f

Example:

• Find all higher-order derivatives of:

  • Polynomial Function: f(x) = 5x⁴ − 12x³ + 12x² − 4x + 1.

Derivative Examples

Example 1:

• Function: f(x) = 2/3x - 1/3

  • Find first, second, and third derivatives.

Example 2:

• Function: f(x) = (2x² + 3)^(2/3)

  • Find first and second derivatives.

Example 3:

• Functions to Derive:

  • (a) f(x) = √(4 − 3x)

  • (b) f(x) = x/(2x + 1)

  • (c) f(x) = (x² + 1)(x − 1)³

  • Find first and second derivatives for each function.

Video Resources

• Suggested Videos on Higher Order Derivatives:

  • Higher Order Derivatives I

  • Higher Order Derivatives II

  • Higher Order Derivatives III

  • Higher Order Derivatives IV

  • Higher Order Derivatives V

  • Higher Order Derivatives VI