一、定义
对于函数 \(f(x)=1,x,x+1,2^x,...\)，我们定义 \(f'(x)=\displaystyle\lim_{\Delta_x\rightarrow0}\frac{f'(x+\Delta_x)-f(x)}{\Delta_x}\)。
二、常见函数求导
-
\(f(x)=1\)
\[\begin{align} f'(x) &=\lim_{\Delta_x\rightarrow0}\frac{f(x+\Delta_x)-f(x)}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{1-1}{\Delta_x}\\ &=0 \end{align} \] -
\(f(x)=x\)
\[\begin{align} f'(x) &=\lim_{\Delta_x\rightarrow0}\frac{f(x+\Delta_x)-f(x)}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{x+\Delta_x-x}{\Delta_x}\\ &=1 \end{align} \] -
\(f(x)=x^2\)
\[\begin{align} f'(x) &=\lim_{\Delta_x\rightarrow0}\frac{f(x+\Delta_x)-f(x)}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{(x+\Delta_x)^2-x^2}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{x^2+2x\Delta_x+\Delta_x^2-x^2}{\Delta_x}\\ &=2x \end{align} \] -
\(f(x)=x^n\)
\[\begin{align} f'(x) &=\lim_{\Delta_x\rightarrow0}\frac{f(x+\Delta_x)-f(x)}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{(x+\Delta_x)^n-x^n}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{\displaystyle\sum_{k=0}^n[{n\choose k}\Delta_x^kx^{n-k}]-x^n}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}\frac{\displaystyle\sum_{k=1}^n[{n\choose k}\Delta_x^kx^{n-k}]}{\Delta_x}\\ &=\lim_{\Delta_x\rightarrow0}{\displaystyle\sum_{k=1}^n[{n\choose k}\Delta_x^{k-1}x^{n-k}]}\\ &={n\choose 1}x^{n-1}\\ &=nx^{n-1} \end{align} \] -
\(f(x)=\sin x\)
\[\begin{align} & \sin(x+\Delta)-\sin x\\ =&\sin x\cos \Delta+\cos x\sin \Delta-\sin x\\ =&\sin x\cdot1+\cos x\sin \Delta -\sin x \end{align} \]则：
\[\begin{align} f'(x)&=\frac{\cos x\sin \Delta}{\Delta}\\ &=\cos x \end{align} \]
三、法则
- \(h(x)=f(x)+g(x)\)\[\begin{align} h'(x)&=\frac{f(x+\Delta)+g(x+\Delta)-f(x)-g(x)}{\Delta}\\ &=f'(x)+g'(x) \end{align} \]
