导数和积分表

\begin{aligned}

1.&f(x)=C,f'(x)=0\

\end{aligned}

\begin{aligned}

2.&f(x)=x^n,f'(x)=nx^{n-1}\

\end{aligned}

\begin{aligned}

3.&f(x)=a^x,f'(x)=ln a\times a^x\

\end{aligned}

\begin{aligned}

4.&f(x)=e^x,f'(x)=e^x\

\end{aligned}

\begin{aligned}

5.&f(x)=log_ax,f'(x)=\frac{1}{x\times ln a}\

&(f(x)=ln x,f'(x)=\frac{1}{x})\

\end{aligned}

\begin{aligned}

6.&f(x)=sin x,f'(x)=cos x\

\end{aligned}

\begin{aligned}

7.&f(x)=cos x,f'(x)=-sin x\

\end{aligned}

\begin{aligned}

8.&f(x)=tan x,f'(x)=\frac{1}{cos^2x}\

\end{aligned}

\begin{aligned}

9.&f(x)=cot x,f'(x)=-\frac{1}{sin^2x}\

\end{aligned}

\begin{aligned}

10.&f(x)=g[h(x)],f'(x)=g'[h(x)]h'(x)\

\end{aligned}

\begin{aligned}

11.&f(x)=g(x)h(x),f'(x)=g'(x)h(x)+g(x)h'(x)\

\end{aligned}

\begin{aligned}

12.&f(x)=\frac{g(x)}{h(x)},f'(x)=\frac{[g'(x)h(x)-g(x)h'(x)]}{h^2(x)}\

\end{aligned}

积分

\begin{aligned}

1.&\int k dx=kx+C\

\end{aligned}

\begin{aligned}

2.&\int k^\mu dx=\frac{x^{\mu +1}}{\mu +1}+C (\mu \not= -1)\

\end{aligned}

\begin{aligned}

3.&\int\frac{dx}{x}=ln|x|+C\

\end{aligned}

\begin{aligned}

4.&\int\frac{dx}{1+x^2}=arctan x+C\

\end{aligned}

\begin{aligned}

5.&\int\frac{dx}{sqrt{1+x^2}}=arcsin x+C\

\end{aligned}

\begin{aligned}

6.&\int cos x dx=sin x+C\

\end{aligned}

\begin{aligned}

7.&\int sin x dx=-cos x+C\

\end{aligned}

\begin{aligned}

8.&\int \frac{dx}{cos^2x}=\int sec^2xdx=tan x+C\

\end{aligned}

\begin{aligned}

9.&\int \frac{dx}{sin^2x}=\int csc^2xdx=-cot x+C\

\end{aligned}

\begin{aligned}

10.&\int sec x tan x dx=sec x+C\

\end{aligned}

\begin{aligned}

11.&\int csc x cot x dx=-csc x+C\

\end{aligned}

\begin{aligned}

12.&\int e^x dx=e^x+C\

\end{aligned}

\begin{aligned}

13.&\int a^x dx=\frac{a^x}{ln a}+C\

\end{aligned}