Unit 3 – Derivative of Algebraic Functions

Algebraic Function

• Algebraic functions are formed by a finite number of operations including addition, subtraction, multiplication, division, raising to powers, and extracting roots.
• Types include polynomial functions and rational functions.
• Polynomial function is defined as ( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 ) where ( a_0, a_1, \ldots, a_n ) are real numbers, ( a_n \neq 0 ), and ( n ) is a non-negative integer.
• An example of a polynomial function is ( f(x) = 2x^3 - 4x^2 + 6x - 8 ).
• Rational functions are expressed as a quotient of two polynomial functions, such as ( h(x) = \frac{x^2 - 25}{x - 4} ).

Differentiation Formulas of Algebraic Functions

• Derivatives help determine the rate of change of functions.
• Key differentiation formulas include:
  • Derivative of a constant: ( \frac{d(c)}{dx} = 0 )
  • Derivative of ( x ): ( \frac{d(x)}{dx} = 1 )
  • Derivative of a power: ( \frac{d(cx^n)}{dx} = cnx^{n-1} )
  • Derivative of a sum/difference: ( \frac{d(u \pm v)}{dx} = \frac{d(u)}{dx} \pm \frac{d(v)}{dx} )
  • Derivative of a product: ( \frac{d(uv)}{dx} = u \frac{d(v)}{dx} + v \frac{d(u)}{dx} )
  • Derivative of a quotient: ( \frac{d\left(\frac{u}{v}\right)}{dx} = \frac{v \frac{d(u)}{dx} - u \frac{d(v)}{dx}}{v^2} )

Application of Differentiation Formulas

• Example of finding derivative:

  • For the function ( y = x^3 - 4x^2 + 6x - 8 ):
    • Use sum rule: ( \frac{d(u \pm v)}{dx} = \frac{d(u)}{dx} \pm \frac{d(v)}{dx} )
    • Apply power rule for each term:
      • ( \frac{d(x^3)}{dx} = 3x^2 )
      • ( \frac{d(-4x^2)}{dx} = -8x )
      • ( \frac{d(6x)}{dx} = 6 )
      • Constant derivative gives ( 0 )

• For composite functions such as ( y = (2x^2 - 3)^2 ):

  • Derivatives may require using the chain rule in addition to previously listed rules.