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 Delicate Derivative Derivation (Posted on 2009-12-18)
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 Comment 2 of 2 |
`Note the similar formula for (a+b)^n andthe nth derivative with respect to x of[(x+1)^2]*[e^(2x)].`
`The nth derivative with respect to x ofe^(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 respectto 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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