फलन `f(x)=(x^100)/(100)+(x^99)/(99)+.....+(x^2)/(2)+x+1` के लिये सिद्ध कीजिए कि `f'(1)=100f'(0)`.

Explanation

यहाँ `f(x)=(x^100)/(100)+(x^99)/(99)+.......+(x^2)/(2)+x+1`
`f'(x) =(d)/(x)[(x^100)/(100)+(x^99)/(99)+........+(x^2)/(2)+x+1]`
`=(1)/(100)(d)/(dx)x^100+(1)/(99) (d)/(dx)x^99+......+(1)/(2)(d)/(dx)x^2+(d)/(dx)x+(d)/(dx)(1)`
`=(1)/(100).100x^99+(1)/(99).99x^98+............+(1)/(2).2x+1+0`
`=x^99+x^98+.......+x+1`
`rArr f'(1)=1+1+........+1+1( 100 "पद तक" ) =100`
तथा `f'(0)=0+0..............+0+1=1`
`therefore f'(1)=100.f'(1)`.