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Delicate Derivative Derivation (Posted on 2009-12-18) Difficulty: 2 of 5
F(x) is a function which is defined as F(x) = (x+1)2 * e2x. The fiftieth derivative of F(x) with respect to x at x=0, i.e. F(50)(0), is expressible in the form k*2n where k is an odd integer and n is a positive integer.

Determine the respective values of k and n.

See The Solution Submitted by K Sengupta    
Rating: 2.0000 (1 votes)

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Solution Solution Comment 2 of 2 |

Note the similar formula for (a+b)^n and
the nth derivative with respect to x of
[(x+1)^2]*[e^(2x)].
The nth derivative with respect to x of
e^(2x), at x=0, is 2^n.
At x=0, the
  oth derivative of (x+1)^2 is 1,
  1st derivative of (x+1)^2 is 2,
  2nd derivative of (x+1)^2 is 2,
  nth derivative of (x+1)^2 is 0
      for n > 2.
Therefore, the 50th derivative with respect
to x of (x+1)^2*e^(2x), at x=0, is
        50!
    ----------- (1)*(2^50)
     0!(50-0)!
        50!
  + ----------- (2)*(2^49)
     1!(50-1)!
        50!
  + ----------- (2)*(2^48)
     2!(50-2)!
  = (1)*(1)*(2^50)+(50)*(2)*(2^49)+(25*49)*(2)*(2^48)
  = (2^49)*(2 + 100 + 1225)
  = 1327*2^49.
Thus, k = 1327 and n = 49.
  

  Posted by Bractals on 2009-12-18 13:51:20
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