Unit 3 – Derivative of Algebraic Functions

Questions and Answers

What is an algebraic function?

An algebraic function is one formed by a finite number of algebraic operations on constants and/or variables.

What is a polynomial function defined by?

f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

What is a rational function?

A function that can be expressed as a quotient of two polynomial functions.

What is the derivative of a constant?

$0$ Signup and view all the answers

What is the derivative of x with respect to x?

1 Signup and view all the answers

What is the formula for the derivative of a power of x?

d/dx (c x^n) = c n x^(n-1) Signup and view all the answers

What is the derivative of a product of two factors?

$d(uv)/dx = u(dv/dx) + v(du/dx)$ Signup and view all the answers

What is the formula for the derivative of a quotient?

d/dx (u/v) = (v(du/dx) - u(dv/dx)) / v^2 Signup and view all the answers

The derivative of a sum/difference of terms is __________.

the sum/difference of their derivatives Signup and view all the answers

What is the common reason students fail to differentiate correctly?

Inadequate knowledge of trigonometry, geometry, and algebra. Signup and view all the answers

Study Notes

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.

Description

This quiz focuses on the concepts related to the derivative of algebraic functions, including polynomial and rational functions. It covers the fundamental operations that define these functions and their significance in calculus. Enhance your understanding of algebraic expressions and their derivatives to excel in this topic.