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

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

Mini-Investigation: Exploring the Derivative of [math]x^n[/math][br][br]Objective:[br]To understand how the derivative of the function [math]f(x)=x^n[/math] changes as we vary the exponent n.[br][br]Procedure:[br]1. Start with the function f(x) = [math]x^n[/math], where n is a positive integer.[br]2. Utilize the definition of the derivative from first principles:[br][br]Investigation Steps:[br]- Slide the value of n from 1 to 10 and record the derivative [math]f'(x)[/math] for each value of[math]n[/math].[br]- Use the applet to expand [math](x+h)^n[/math] and [math]x^n[/math] to observe the patterns in the coefficients of the resulting polynomial.[br][br]Analysis:[br]1. Observe the coefficients of the terms in the expanded form of [math](x+h)^n-x^n[/math].[br]2. Note how the h term disappears as we take the limit as h approaches zero, leaving us with the derivative function.[br]3. Relate the coefficients of the derivative to the original exponent [math]n[/math].[br][br]Conclusions:[br]- Formulate a general rule for the derivative of [math]x^n[/math] based on the patterns observed.[br]- Discuss how the power rule for differentiation, [math]f'(x)=nx^{\left(n-1\right)}[/math], arises from the investigation.[br]- Reflect on the efficiency of using first principles versus the power rule for finding derivatives.[br]

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.[br]