公式
\((u \pm v)' = u' \pm v'\)
\((uv)' = u'v + uv'\)
\((\frac{u}{v})' = \frac{u'v - v'u}{v^2}\)
证明（导数的定义）
\((u \pm v)' = \lim_{\Delta x \to 0}\frac{(u(x+\Delta x)\pm v(x+\Delta x)) - (u(x)\pm v(x))}{\Delta x}\)
\(=\lim_{\Delta x \to 0}\frac{(u(x+\Delta x) - u(x)) \pm (v(x+\Delta x) - v(x))}{\Delta x}\)
\(=u' \pm v'\)
\((uv)' = \lim_{\Delta x \to 0}\frac{u(x+\Delta x)\cdot v(x+\Delta x) - u(x)\cdot v(x)}{\Delta x}\)
\(=\lim_{\Delta x \to 0}\frac{v(x+\Delta x)\cdot(u(x+\Delta x) - u(x)) + u(x)\cdot(v(x+\Delta x) - v(x))}{\Delta x}\)
\(=u'v + uv'\)
标签：uv,frac,cdot,lim,微积分,四则运算,Delta,求导,pm From： https://www.cnblogs.com/J-12045/p/18554061\((\frac{u}{v})' = \lim_{\Delta x \to 0}\frac{\frac{u(x+\Delta x)} {v(x+\Delta x)} - \frac{u(x)}{v(x)}}{\Delta x}\)
\(=\lim_{\Delta x \to 0}\frac{u(x+\Delta x)\cdot v(x) - u(x)\cdot v(x+\Delta x)}{\Delta x \cdot v(x+\Delta x) \cdot v(x)}\)
\(=\lim_{\Delta x \to 0}\frac{v(x)\cdot(u(x+\Delta x) - u(x)) - u(x)\cdot(v(x+\Delta x) - v(x))}{\Delta x \cdot v(x+\Delta x) \cdot v(x)}\)
\(= \frac{u'v - v'u}{v^2}\)