This document outlines the algebraic techniques of differentiation, including the derivatives of constants, power functions, sums and differences of functions, products of functions, and quotients of functions. It provides examples of applying the differentiation rules to find the derivatives of various functions.
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A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
Presntation for the post of lecturer in MathematicsKifayat Ullah
This talk was given during the recruitment process for the post of Lecturer in Mathematics, at University of Science and technology Bannu, on the topic of "Comparison of Reiman integration and lesbegue integration theories".
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It is a powerpoint presentation that discusses about the lesson or topic of Derivatives and Differentiation Rules. It also encompasses some formulas, definitions and examples regarding the said topic.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
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1. The definition of derivative can lead us to the derivation of the rules of differentiation or the algebraic
techniques of differentiation.
ALGEBRAIC TEHCNIQUES OF DIFFERENTIATION
ALGEBRAIC TEHCNIQUES OF
DIFFERENTIATION
EXAMPLES
A. Derivative of a Constant
If 𝑓(𝑥) = 𝑘, where 𝑘 is a real
number, then 𝑓′(𝑥) = 0
𝑓(𝑥) = 3
𝒇′(𝒙) = 𝟎
B. Derivative of a Power Function
(Power Rule)
If 𝑓(𝑥) = 𝑥𝑛
, where 𝑛 is a real
number, then 𝑓′(𝑥) = 𝑛𝑥𝑛−1
𝑓(𝑥) = 𝑥2
𝑓(𝑥) = 2𝑥2−1
𝒇(𝒙) = 𝟐𝒙
C. Derivative of the Sum or
Difference of Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) = 𝑓(𝑥) ±
𝑔(𝑥) then 𝑝′(𝑥) = 𝑓′(𝑥) ± 𝑔′(𝑥)
𝑓(𝑥) = 3𝑥4
− 2𝑥3
+ 4𝑥2
− 12𝑥 − 5
𝑓′(𝑥) = (4)(3)(𝑥4−1) − (3)(2)(𝑥3−1) + (2)(4)(𝑥1−1) − 0
𝒇′(𝒙) = 𝟏𝟐𝒙𝟑
− 𝟔𝒙𝟐
+ 𝟖𝒙
D. Derivative of the Product of Two
Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) = 𝑓(𝑥) ∙
𝑔(𝑥) then 𝑝′(𝑥) = [𝑓(𝑥) ∙ 𝑔′(𝑥)] ∙
[𝑓′(𝑥)𝑔(𝑥)]
𝑓(𝑥) = (2𝑥2
+ 1)(4𝑥2
+ 2)
𝑓′(𝑥) = (2𝑥2
+ 1)[(4)(2)(𝑥2−1)] + [(2)(2)(𝑥2−1)](4𝑥2
+ 2)
𝑓′(𝑥) = (2𝑥2
+ 1)(8𝑥) + (4𝑥)(4𝑥2
+ 2)
𝑓′(𝑥) = 16𝑥2
+ 8𝑥 + 16𝑥2
+ 8𝑥
𝑓′(𝑥) = 32𝑥2
+ 16𝑥
E. Derivative of the Quotient of
Two Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) =
𝑓(𝑥)
𝑔(𝑥)
where 𝑔(𝑥) ≠ 0, then
𝑝′(𝑥) =
𝑔(𝑥)𝑓′(𝑥)−𝑓(𝑥)𝑔′(𝑥)
[𝑔(𝑥)]2
𝑓(𝑥) =
2𝑥 − 3
4𝑥 + 1
𝑓′(𝑥) =
(4𝑥 + 1)(2) − (2𝑥 − 3)(4)
(4𝑥 + 1)2
𝑓′(𝑥) =
(8𝑥 + 2) − (8𝑥 − 12)
(4𝑥 + 1)(4𝑥 = 1)
𝑓′(𝑥) =
14
16𝑥2 + 8𝑥 + 1
![The definition of derivative can lead us to the derivation of the rules of differentiation or the algebraic
techniques of differentiation.
ALGEBRAIC TEHCNIQUES OF DIFFERENTIATION
ALGEBRAIC TEHCNIQUES OF
DIFFERENTIATION
EXAMPLES
A. Derivative of a Constant
If 𝑓(𝑥) = 𝑘, where 𝑘 is a real
number, then 𝑓′(𝑥) = 0
𝑓(𝑥) = 3
𝒇′(𝒙) = 𝟎
B. Derivative of a Power Function
(Power Rule)
If 𝑓(𝑥) = 𝑥𝑛
, where 𝑛 is a real
number, then 𝑓′(𝑥) = 𝑛𝑥𝑛−1
𝑓(𝑥) = 𝑥2
𝑓(𝑥) = 2𝑥2−1
𝒇(𝒙) = 𝟐𝒙
C. Derivative of the Sum or
Difference of Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) = 𝑓(𝑥) ±
𝑔(𝑥) then 𝑝′(𝑥) = 𝑓′(𝑥) ± 𝑔′(𝑥)
𝑓(𝑥) = 3𝑥4
− 2𝑥3
+ 4𝑥2
− 12𝑥 − 5
𝑓′(𝑥) = (4)(3)(𝑥4−1) − (3)(2)(𝑥3−1) + (2)(4)(𝑥1−1) − 0
𝒇′(𝒙) = 𝟏𝟐𝒙𝟑
− 𝟔𝒙𝟐
+ 𝟖𝒙
D. Derivative of the Product of Two
Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) = 𝑓(𝑥) ∙
𝑔(𝑥) then 𝑝′(𝑥) = [𝑓(𝑥) ∙ 𝑔′(𝑥)] ∙
[𝑓′(𝑥)𝑔(𝑥)]
𝑓(𝑥) = (2𝑥2
+ 1)(4𝑥2
+ 2)
𝑓′(𝑥) = (2𝑥2
+ 1)[(4)(2)(𝑥2−1)] + [(2)(2)(𝑥2−1)](4𝑥2
+ 2)
𝑓′(𝑥) = (2𝑥2
+ 1)(8𝑥) + (4𝑥)(4𝑥2
+ 2)
𝑓′(𝑥) = 16𝑥2
+ 8𝑥 + 16𝑥2
+ 8𝑥
𝑓′(𝑥) = 32𝑥2
+ 16𝑥
E. Derivative of the Quotient of
Two Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) =
𝑓(𝑥)
𝑔(𝑥)
where 𝑔(𝑥) ≠ 0, then
𝑝′(𝑥) =
𝑔(𝑥)𝑓′(𝑥)−𝑓(𝑥)𝑔′(𝑥)
[𝑔(𝑥)]2
𝑓(𝑥) =
2𝑥 − 3
4𝑥 + 1
𝑓′(𝑥) =
(4𝑥 + 1)(2) − (2𝑥 − 3)(4)
(4𝑥 + 1)2
𝑓′(𝑥) =
(8𝑥 + 2) − (8𝑥 − 12)
(4𝑥 + 1)(4𝑥 = 1)
𝑓′(𝑥) =
14
16𝑥2 + 8𝑥 + 1](https://image.slidesharecdn.com/thealgebraictechniquesmodule4-210225034112/85/The-algebraic-techniques-module4-1-320.jpg)
