华中师范大学：《数学分析》课程PPT教学课件（讲稿）第二章（2.2.2）导数的定义

如果函数=(x)及v=x)在点x具有导数,那么它们的积在 点x也具有导数,并且 Lu(x)v(x)=u'(x)v(x+u(x)y'() 证明由导数的定义, [(x)·v(x)]=lim u(x+hv(x+h)-u(xv(x h→>0 h =lim[(x+b)(x+h)-(x(x+h)+(x)(x+)-(x)(x) h→>0 (x+h)-l(x) v(x+h)+u(x) v(x+h)-v(x) h h lim u(r+ ch-u(x) h im v(x+h)+u(x). lim (x+h)-v(x) h→>0 h→>0 h->0 h =l(x)(x)+(x)v(x) 上页 下页

上页 返回 下页 如果函数u=u(x)及v=v(x)在点x具有导数那么它们的积在 点x也具有导数 并且 [u(x)v(x)]=u(x)v(x)+u(x)v(x)  h u x h v x h u x v x u x v x h ( ) ( ) ( ) ( ) [ ( ) ( )] lim 0 + + −   = → [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] 1 lim 0 u x h v x h u x v x h u x v x h u x v x h h = + + − + + + − →  + −  + +   + − = → h v x h v x v x h u x h u x h u x h ( ) ( ) ( ) ( ) ( ) ( ) lim 0 h v x h v x v x h u x h u x h u x h h h ( ) ( ) lim ( ) ( ) lim ( ) ( ) lim 0 0 0 + −  + +  + − = → → → =u(x)v(x)+u(x)v(x) 证明 由导数的定义