Derivative of x^n

Keywords

Derivative	微分	미분	导数
Function	関数	함수	函数
Exponent	指数	지수	指数
First principles	基本原理	기본 원리	基本原理
Polynomial	多項式	다항식	多项式
Coefficients	係数	계수	系数
Limit	極限	극한	极限
h term	h項	h 항	h项
Power rule	累乗法則	거듭제곱 법칙	幂规则
Differentiation	微分法	미분법	微分
General rule	一般規則	일반 규칙	一般规则
Efficiency	効率性	효율성	效率
n from 1 to 10	1から10までのn	n을 1부터 10까지	n从1到10
Expanded form	展開形	전개된 형태	展开式
Derivative function	導関数	미분 함수	导函数
Original exponent	元の指数	원래의 지수	原指数
Using first principles	基本原理を用いて	기본 원칙 사용하기	使用基本原理
Finding derivatives	導関数の求め方	미분 찾기	求导数

Factual Questions	Debatable Questions	Conceptual Questions
1. What is the power rule for differentiation as stated in the document?	1. Is the power rule always the most efficient method for finding derivatives of polynomials?	1. How does the concept of limits underpin the definition of a derivative?
2. According to the procedure, what does represent?	2. Can the process used in first principles be considered more educational than using the power rule?	2. Why does the term disappear when taking the limit as approaches zero?
3. What pattern in coefficients is observed when expanding ?	3. Is the differentiation of functions of the form where is a positive integer inherently simpler than other types of functions?	3. In what way do the coefficients of a polynomial function relate to its derivative?
4. How is the derivative function calculated from first principles for ?	4. Should the first principles method be taught before introducing rules like the power rule?	4. What does it mean for a function to be differentiable?
5. What is the general rule formulated for the derivative of based on the mini-investigation?	5. Could there be a better approach to teaching derivatives than the methods currently used?	5. How does differentiating a polynomial affect its degree and the nature of its graph?

Mini-Investigation: Exploring the Derivative of

Objective: To understand how the derivative of the function

changes as we vary the exponent n. Procedure: 1. Start with the function f(x) =

, where n is a positive integer. 2. Utilize the definition of the derivative from first principles: Investigation Steps: - Slide the value of n from 1 to 10 and record the derivative

for each value of

. - Use the applet to expand

and

to observe the patterns in the coefficients of the resulting polynomial. Analysis: 1. Observe the coefficients of the terms in the expanded form of

. 2. Note how the h term disappears as we take the limit as h approaches zero, leaving us with the derivative function. 3. Relate the coefficients of the derivative to the original exponent

. Conclusions: - Formulate a general rule for the derivative of

based on the patterns observed. - Discuss how the power rule for differentiation,

, arises from the investigation. - Reflect on the efficiency of using first principles versus the power rule for finding derivatives.

Part 2 - The power rule - Differentiating polynomials

In practice it's not necesary to always differentiate from first principles. The power rule can be used directly, so we are able to differentiate any polynomials quickly. Watch the following video.

1. What is the derivative of with respect to ?

Select all that apply