高等数学_导数公式默诵练习-题解-3

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 1(x)1'a(x(x)-l(x)y(x) v( 证明由导数的定义, u(x+h) u(x) u(x)=lim v(x+h)v(x) lim u(x+h)v(x-u(xv(x+h) h→>0 h h->0 v(x+hv(xh lim lu(x+h)-u(x)j(x)=u(x)[v(x+h)-v(x h→>0 v(x+hv(h l(x+h)-(x) v(x)u(x) v(x+h)-v(x h→>0 v(x+hv(x) u'(xv(x)=u(xv(x) 上页 返回 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 证明 由导数的定义 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 v x u x v x u x v x v x u x  −  =         v x h v x h u x h v x u x v x h h v x u x v x h u x h v x u x h h ( ) ( ) ( ) ( ) ( ) ( ) lim ( ) ( ) ( ) ( ) lim ( ) ( ) 0 0 + + − + = − + + =        → → v x h v x h u x h u x v x u x v x h v x h ( ) ( ) [ ( ) ( )] ( ) ( )[ ( ) ( )] lim 0 + + − − + − = → ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) lim 0 v x h v x h v x h v x v x u x h u x h u x h + + − − + − = → ( ) ( ) ( ) ( ) ( ) 2 v x u  x v x −u x v  x =  v x h v x h u x h v x u x v x h h v x u x v x h u x h v x u x h h ( ) ( ) ( ) ( ) ( ) ( ) lim ( ) ( ) ( ) ( ) lim ( ) ( ) 0 0 + + − + = − + + =        → →