导数和微积分(蒯)

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\
\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)=ex\
\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}