微积分_例题解-3

例3.求正弦函数与余弦函数的m阶导数 解: y= x, y=cosx=sin(x+a) y=co(x+a)=sI(x++2)=Si(x+2.2), y=coS(x+2.7t 2=sn(x+3. y()=cos(x+3/)=sin(x+4 般地,可得 A n)=sin(x+n.), EJ(sin x)n)=sin(x+n) 用类似方法,可得(c0sx)y)=cos(x+n·x) 上页 结束 下页

上页 结束 下页 解 例3 求正弦函数与余弦函数的n阶导数 y=sin x ) 2 cos sin(  y  = x= x+  ) 2 ) sin( 2 2 2 ) sin( 2 cos(     y  = x+ = x+ + = x+   ) 2 ) sin( 3 2 cos( 2   y  = x+  = x+   ) 2 ) sin( 4 2 cos( 3 (4)   y = x+  = x+   一般地 可得 ) 2 sin( ( )  y = x+n n  即 ) 2 (sin ) sin( ( )  x = x+n n  用类似方法 可得 ) 2 (cos ) cos( ( )  x = x+n n  ) 2 cos sin(  y  = x= x+  ) 2 ) sin( 2 2 2 ) sin( 2 cos(     y  = x+ = x+ + = x+  )  2 ) sin( 2 2 2 ) sin( 2 cos(     y  = x+ = x+ + = x+   ) 2 ) sin( 3 2 cos( 2   y  = x+  = x+   ) 2 ) sin( 4 2 cos( 3 (4)   y = x+  = x+   ) 2 sin( ( )  y = x+n n  即 ) 2 (sin ) sin( ( )  x = x+n n 