How to apply Leibnitz theorem in any equation

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I want to know that how to apply leibnitz theorem

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Mathematics Lala Soni 4 months 1 Answer 559 views 0

Answer ( 1 )

  1. Hi, The nth derivative of the product of two functions.

    Leibnitz gave the theorem to find nth derivative of the product of two functions

    According to this theorem, if u and v are two functions of x, then the nth derivative of the product of u and v will be given by

    (uv)n = uovn + nC1u1vn-1 + nC2u2vn-2 +….. nCnunvo

    For applying this theorem to find the nth derivative, one needs to figure out the functions properly.
    Following steps can be helpful:

    1. Look at the function properly and transform the function into such a form that direct nth differential coefficient of different standard functions can be used.
    2. Remember the standard nth differential coefficient of different standard functions like cos(ax + b), sin(ax + b), abx, (ax+b), eaxsin(bx+c), eaxcos(bx+c) etc.
    3. nth differential coefficient of some functions are as follows

    Dn (xm) = m(m-1)(m-2)…..(m-n+1)xm-n

    Dn (log x) = [(-1)n-1(n-1)!]/xn

    Dn (sin(ax+b)) = ansin[ax + b + nπ/2 ]

    Dn (cos(ax+b)) = ancos[ax + b + nπ/2 ]

    for example, nth differential of function sinaxcosbx

    Let, y = sinaxcosbx = 1/2 [2 sinaxcosbx] = 1/2 [2 sin (a+b)x + sin (a-b)x]

    We know, Dn (sin(ax+b)) = ansin[ax + b + nπ/2 ]

    So, yn = 1/2 [ (a+b)nsin{(a+b)x+ nπ/2} + (a-b)nsin{(a-b)x + nπ/2} ]

    Best answer

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