## 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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## Answer ( 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}= u_{o}v_{n}+_{n}C^{1}u_{1}v_{n-1}+_{n}C^{2}u_{2}v_{n-2}+….._{n}C^{n}u_{n}v_{o}For applying this theorem to find the nth derivative, one needs to figure out the functions properly.

Following steps can be helpful:

^{bx}, (ax+b), e^{ax}sin(bx+c), e^{ax}cos(bx+c) etc.D

^{n}(x^{m}) = m(m-1)(m-2)…..(m-n+1)x^{m-n}D

^{n }(log x) = [(-1)^{n-1}(n-1)!]/x^{n}D

^{n}(sin(ax+b)) = a^{n}sin[ax + b + nπ/2 ]D

^{n}(cos(ax+b)) = a^{n}cos[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, D

^{n}(sin(ax+b)) = a^{n}sin[ax + b + nπ/2 ]So, y

^{n}= 1/2 [ (a+b)^{n}sin{(a+b)x+ nπ/2} + (a-b)^{n}sin{(a-b)x + nπ/2} ]