Question

Find the nth derivative y=xsquare sin3x using leibinitz theorem

Answer 1

Leibniz Formula for  nth derivative of a product of two functions u and v of x is:

$u^n_x=n^{th}derivative\ of\ u\ wrt\ x.\\\\(uv)^n_x=u^n_x\ v+{}^nC_1\ u^{n-1}_x\ v^1_x + {}^nC_2\ u^{n-2}_x\ v^2_x +\\. \ \ \ \ .\ . \ . \ +{}^nC_{n-1}\ u^1_x\ v^{n-1}_x + u\ v^n_x\\\\u=sin3x,\ v=x^2\\\\v^1_x=2x,\ v^2_x=2,\ \ v^r_x=0,\ for\ r\ \textgreater \ 2\\\\u^1_x=3cos3x,\ u^2_x=-3^2sin3x,\ u^3_x=-3^3cos3x,\\\\u^4_x=3^4sin3x,\ \ u^5=3^5cos3x,\ u^6_x=-3^6sin3x,\ u^7_x=-3^7cos3x\\\\u^r_x=3^r\ cos[3x+(r-1)\frac{\pi}{2}]$

 

[tex] (sin3x *x^2)^n_x=3^n\ cos[3x+(n-1)\frac{\pi}{2}]\ x^2+\\\\. \ \ \ \ \ +n\ 3^{n-1}\ cos[3x+(n-2)\frac{\pi}{2}]\ 2x+3^{n-2}\ cos[3x+(n-3)\frac{\pi}{2}]\ 2[/tex]