Calculus - Basic Concepts of Calculus

Enterprise 2023-08-12 21:23:24 views: null

Learn now: https://edu.csdn.net/course/play/26262/326625?utm_source=blogtoedu

Derivative:

\lim_{\triangle x \to 0}\frac{\triangle f(x)}{\triangle x} =\frac{f(x_{0}+\triangle x)-f(x_{0}) }{\triangle x}=f'(x_{0})

 

Take the derivative of x raised to the nth power:

Q: f(x)=x^{n}, pleasef'(x_0)

A：

\begin{aligned} x_0+\triangle x^n &=(x_0+\triangle x)(x_0+\triangle x)...(x_0+\triangle x) \\ &=x_0^n+nx_0^{n-1}\triangle x+O((\triangle x)^2)\\ \end{aligned}

\begin{aligned} f'(x_0)&=\lim_{\triangle x \to 0}\frac{\triangle f(x)}{\triangle x}\\ &=\lim_{\triangle x \to 0}\frac{\triangle f(x_0+\triangle x)-f(x_0)}{\triangle x}\\ &=\lim_{\triangle x \to 0}\frac{x_0^n+nx_0^{n-1}\triangle x+O((\triangle x)^2)-x_0^n}{\triangle x}\\ &=\lim_{\triangle x \to 0}nx^{n-1}+O(\triangle x)\\ &=nx^{n-1} \end{aligned}

 

(LaTeX formula online editor https://www.latexlive.com/ )

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Origin blog.csdn.net/Dugege007/article/details/105471901
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