导数的四则运算

castelu

定理1　若函数$u\left( x \right)$和$v\left( x \right)$在点$x_0$可导，则函数$f\left( x \right)=u\left( x \right) \pm v\left( x \right)$在点$x_0$也可导，且
$$f'\left( x_0 \right)=u'\left( x_0 \right) \pm v'\left( x_0 \right)$$

定理2　若函数$u\left( x \right)$和$v\left( x \right)$在点$x_0$可导，则函数$f\left( x \right)=u\left( x \right)v\left( x \right)$在点$x_0$也可导，且
$$f'\left( x_0 \right)=u'\left( x_0 \right)v\left( x_0 \right)+u\left( x_0 \right)v'\left( x_0 \right)$$

推论　若函数$v\left( x \right)$在点$x_0$可导，$c$为常数，则
$$\left( cv\left( x \right) \right)'_{x=x_0}=cv'\left( x_0 \right)$$

定理3　若函数$u\left( x \right)$和$v\left( x \right)$在点$x_0$都可导，且$v\left( x_0 \right) \ne 0$，则$f\left( x \right)=\frac{u\left( x \right)}{v\left( x \right)}$在点$x_0$也可导，且
$$f'\left( x_0 \right)=\frac{u'\left( x_0 \right)v\left( x_0 \right)-u\left( x_0 \right)v'\left( x_0 \right)}{\left[ v\left( x_0 \right) \right]^2}$$