# 求导加法法则

## 求导加法法则

(u\pm v)^\prime=u^\prime\pm v^\prime

[u(x)+v(x)]^\prime=u^\prime(x)+v^\prime(x)\\ [u(x)-v(x)]^\prime=u^\prime(x)-v^\prime(x)

\displaystyle [u(x)+v(x)]^\prime=\lim\limits_{\Delta x\to0}\frac{[u(x+\Delta x)+v(x+\Delta x)]-[u(x)+v(x)]}{\Delta x}\\ {}\\ =\lim\limits_{\Delta x\to0}\frac{u(x+\Delta x)-u(x)}{\Delta x}+\lim\limits_{\Delta x\to0}\frac{v(x+\Delta x)-v(x)}{\Delta x}\\ {}\\ =u^\prime(x)+v^\prime(x)

\displaystyle [u(x)-v(x)]^\prime=\lim\limits_{\Delta x\to0}\frac{[u(x+\Delta x)-v(x+\Delta x)]-[u(x)-v(x)]}{\Delta x}\\ {}\\ =\lim\limits_{\Delta x\to0}\frac{u(x+\Delta x)-u(x)}{\Delta x}-\lim\limits_{\Delta x\to0}\frac{v(x+\Delta x)-v(x)}{\Delta x}\\ {}\\ =u^\prime(x)-v^\prime(x)

\displaystyle f(x)=\sin x+\cos x\\{}\\ f^\prime(x)=(\sin x+\cos x)^\prime=(\sin x)^\prime+(\cos x)^\prime\\{}\\ f^\prime=\cos x-\sin x\\{}\\ f(x)=\cos x+x^6\Longrightarrow f^\prime=-\sin x+6x^5\\{}\\ f(x)=\frac{1}{x}-x^2\Longrightarrow f^\prime=-\frac{1}{x^2}-2x\\{}\\ f(x)=\sin x-\frac{2}{x}\Longrightarrow f^\prime=\cos x+\frac{2}{x^2}