Home
Class 12
MATHS
If f(x)=(x-1)^(100)(x-2)^(2(99))(x-3)^(3...

If `f(x)=(x-1)^(100)(x-2)^(2(99))(x-3)^(3(98))…(x-100)^(100),` then the value of `(f'(101))/(f(101))` is Option 1: 5050 Option 2: 2575 Option 3: 3030 Option 4: 1250

A

5050

B

2575

C

3030

D

1250

Text Solution

Verified by Experts

The correct Answer is:
A
`f(x)=prod_(i=l)^(100)(x-i)^(i(101-i))`
Taking log both sides, we get
`log(f(x))=sum_(i=l)^(100)i(101-i)log(x-i)`
Differentiating w.r.t. x, we get
`(1)/(f(x)).f'(x)=sum_(i=l)^(100)(i(101-i))/((x-i))`
`rArr" "(f'(100))/(f(101))=sum_(i=l)^(100)((101-i)/(101-i))=5050`
Promotional Banner

Topper's Solved these Questions

  • METHODS OF DIFFERENTIATION

    CENGAGE|Exercise Multiple Correct Answer Type|7 Videos

  • MATRICES

    CENGAGE|Exercise Multiple Correct Answer|7 Videos

  • METHODS OF DIFFERETIATION

    CENGAGE|Exercise Question Bank|13 Videos

Similar Questions

Explore conceptually related problems

"If "f(x)=(x-1)^(4)(x-2)^(3)(x-3)^(2)(x-4), then the value of f'''(1)+f''(2)+f'(3)+f'(4) equals

If g(x)=(4cos^4x-2cos2x-1/2cos4x-x^7)^(1/7) then the value of g(g(100)) is equal to Option 1: -1 Option 2: 0 Option 3: 1 Option 4: 100

If t(1+x^2)=x and x^2+t^2=y, then at x=2 the value of (d y)/(d x) is equal to Option 1. 24/5 Option 2: 101/125 Option 3: 488/155 Option 4: 358/125

Let f(x)=sin^(-1)((2x)/(1+x^(2))) and g(x)=cos^(-1)((x^(2)-1)/(x^(2)+1)) . Then tha value of f(10)-g(100) is equal to

If f(x)=prod_(n=1)^(100)(x-n)^(n(101-n)) ; then (f(101))/(f'(101))=

"If " int e^(x^(3)+x^(2)-1)(3x^(4)+2x^(3)+2x)dx=f(x)+C, " then the value of " f(1)xxf(-1) " is"-.

If f(x) =(3(Inx)^(2)+(In x)^(4))/(x(1+(In x)^(2)-(Inx)^(3))^(2))dx and f(1) = 0, then the value of |3f(e^(2))| is equal to

If (x^2+x−2)/(x+3) 1) f(x) then find the value of lim_(x->1) f(x)

"Let "f(x)=((2^(x)+2^(-x))sin x sqrt(tan^(-1)(x^(2)-x+1)))/((7x^(2)+3x+1)^(3)) . Then find the value of f'(0).

If f(x)=2x^(2)+3x-5 , then prove that f'(0)+3f'(-1)=0