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AP EAMCET2010MathematicsDifferentiation

If f(x)=( x)( 2 x) ( n x) then f^ (x)+ _ r=1 ^n(r r x) f(x) is equal to

Options

  1. Af(x)
  2. B0
  3. C-f(x)
  4. D2 f(x)

Correct answer

B. 0

Step-by-step solution

f(x)=( x)( 2 x) ( n x)f^ (x)=- x 2 x n x+ x d d x 2 x 3 x n x f^ (x)=- x 2 x n x+ x[-2 2 x 3 x n x .+ 2 x d d x 3 x 4 x n x ]f^ (x) -( x 2 x n x)-(2 x 2 x 3 x n x)+ x 2 x d d x ( 3 x 4 x n x)f^ (x) -( x 2 x n x)-(2 x 2 x n x)-(3 x 2 x 3 x n x)-(n x 2 x n x) So, f^ (x)+ _ r=1 ^n(r r x) f(x)=f^ (x)+ x+2 2 x+3 3 x+ +n n x f(x)=f^ (x)+f(x) x+2 f(x) 2 x+ +n f(x) n x=f^ (x)+[( x 2 x n x)+(2 x 2 x n x)+ +(n x 2 x n x)]=f^ (x)-f^ (x) 0 Hence, f^ (x)+ _ r=1 ^n(r r x) f(x)=0

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