导数和微积分(蒯)

导数和积分表

egin{aligned}
1.&f(x)=C,f'(x)=0
end{aligned}
egin{aligned}
2.&f(x)=x^n,f'(x)=nx^{n-1}
end{aligned}
egin{aligned}
3.&f(x)=a^x,f'(x)=ln a imes a^x
end{aligned}
egin{aligned}
4.&f(x)=e^x,f'(x)=e^x
end{aligned}
egin{aligned}
5.&f(x)=log_ax,f'(x)=frac{1}{x imes ln a}
&(f(x)=ln x,f'(x)=frac{1}{x})
end{aligned}
egin{aligned}
6.&f(x)=sin x,f'(x)=cos x
end{aligned}
egin{aligned}
7.&f(x)=cos x,f'(x)=-sin x
end{aligned}
egin{aligned}
8.&f(x)=tan x,f'(x)=frac{1}{cos^2x}
end{aligned}
egin{aligned}
9.&f(x)=cot x,f'(x)=-frac{1}{sin^2x}
end{aligned}
egin{aligned}
10.&f(x)=g[h(x)],f'(x)=g'[h(x)]h'(x)
end{aligned}
egin{aligned}
11.&f(x)=g(x)h(x),f'(x)=g'(x)h(x)+g(x)h'(x)
end{aligned}
egin{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

egin{aligned}
1.&int k dx=kx+C
end{aligned}
egin{aligned}
2.&int k^mu dx=frac{x^{mu +1}}{mu +1}+C (mu ot= -1)
end{aligned}
egin{aligned}
3.&intfrac{dx}{x}=ln|x|+C
end{aligned}
egin{aligned}
4.&intfrac{dx}{1+x^2}=arctan x+C
end{aligned}
egin{aligned}
5.&intfrac{dx}{sqrt{1+x^2}}=arcsin x+C
end{aligned}
egin{aligned}
6.&int cos x dx=sin x+C
end{aligned}
egin{aligned}
7.&int sin x dx=-cos x+C
end{aligned}
egin{aligned}
8.&int frac{dx}{cos^2x}=int sec^2xdx=tan x+C
end{aligned}
egin{aligned}
9.&int frac{dx}{sin^2x}=int csc^2xdx=-cot x+C
end{aligned}
egin{aligned}
10.&int sec x tan x dx=sec x+C
end{aligned}
egin{aligned}
11.&int csc x cot x dx=-csc x+C
end{aligned}
egin{aligned}
12.&int e^x dx=e^x+C
end{aligned}
egin{aligned}
13.&int a^x dx=frac{a^x}{ln a}+C
end{aligned}