导数与积分

导数

\[f'(x)=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

有用的导数基本公式：

\[\begin{aligned} f(x)=e^x&\implies f'(x)=e^x\\ f(x)=a&\implies f'(x)=0\\ f(x)=\ln a&\implies f'(x)=\frac{1}{x}\\ f(x)=ax^t&\implies f'(x)=atx^{t-1}\\ (f(x)+g(x))'&\implies f'(x)+g'(x)\\ (f(g(x)))'&\implies f'(g(x))\times g'(x) \end{aligned} \]

\[f(x)=\sum_{i=0}^n a_ix^i\implies f'(x)=\sum_{i=0}^{n-1}(i+1)a_{i+1}x^i \]

积分

多项式 \(f(x)=\sum_{i=0}^na_ix^i\) 积分：

\[\sum_{i=1}^{n+1} \frac{a_{i-1}x^i}{i} \]