數論基礎——求導

辜铜星發表於2024-09-09

一、定義

對於函式 \(f(x)=1,x,x+1,2^x,...\),我們定義 \(f'(x)=\displaystyle\lim_{\Delta_x\rightarrow0}\frac{f'(x+\Delta_x)-f(x)}{\Delta_x}\)

二、常見函式求導

  1. \(f(x)=1\)

    \[\begin{align} f'(x) &=\lim_{\Delta_x\rightarrow0}\frac{f(x+\Delta_x)-f(x)}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{1-1}{\Delta_x}\\ &=0 \end{align} \]

  2. \(f(x)=x\)

    \[\begin{align} f'(x) &=\lim_{\Delta_x\rightarrow0}\frac{f(x+\Delta_x)-f(x)}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{x+\Delta_x-x}{\Delta_x}\\ &=1 \end{align} \]

  3. \(f(x)=x^2\)

    \[\begin{align} f'(x) &=\lim_{\Delta_x\rightarrow0}\frac{f(x+\Delta_x)-f(x)}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{(x+\Delta_x)^2-x^2}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{x^2+2x\Delta_x+\Delta_x^2-x^2}{\Delta_x}\\ &=2x \end{align} \]

  4. \(f(x)=x^n\)

    \[\begin{align} f'(x) &=\lim_{\Delta_x\rightarrow0}\frac{f(x+\Delta_x)-f(x)}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{(x+\Delta_x)^n-x^n}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{\displaystyle\sum_{k=0}^n[{n\choose k}\Delta_x^kx^{n-k}]-x^n}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{\displaystyle\sum_{k=1}^n[{n\choose k}\Delta_x^kx^{n-k}]}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}{\displaystyle\sum_{k=1}^n[{n\choose k}\Delta_x^{k-1}x^{n-k}]}\\ &={n\choose 1}x^{n-1}\\ &=nx^{n-1} \end{align} \]

  5. \(f(x)=\sin x\)

    \[\begin{align} & \sin(x+\Delta)-\sin x\\ =&\sin x\cos \Delta+\cos x\sin \Delta-\sin x\\ =&\sin x\cdot1+\cos x\sin \Delta -\sin x \end{align} \]

    則:

    \[\begin{align} f'(x)&=\frac{\cos x\sin \Delta}{\Delta}\\ &=\cos x \end{align} \]

三、法則

  1. \(h(x)=f(x)+g(x)\)

    \[\begin{align} h'(x)&=\frac{f(x+\Delta)+g(x+\Delta)-f(x)-g(x)}{\Delta}\\ &=f'(x)+g'(x) \end{align} \]

相關文章