## How to apply Leibnitz theorem in any equation

Question

I want to know that how to apply leibnitz theorem

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Mathematics Lala Soni 4 months 1 Answer 559 views 0

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.

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} ]