Question

if y = x² sinx then prove that : d^ny/dx^n = (x²-n²+n) sin(x+nπ/2) -2nx cos(x+nπ/2)
​

Answer 1

Answer:

Given

y=x

2

sinx

Using

dx

d(u .v)

=v

dx

du

+u

dx

dv

Differentiating on both sides

⟹

dx

dy

=

dx

d

(x

2

sinx)

⟹

dx

dy

=sinx

dx

d

(x

2

)+x

2

dx

d

(sinx)

⟹

dx

dy

=2xsinx+x

2

cosx

Answer 2

Answer:

To get the nth derivative of the product of 2 functions, we can use Leibniz rule of successive differentiation. It says, if  u  and  v  are differentiable functions of  x , then the  n th derivative of  uv  is given by the following expression,

dndxnu×v=nC0unv+nC1un−1v1+nC2un−2v2+⋯+nCrun−rvr+⋯+nCnuvn  

We conveniently choose the second function as  x2 , because we know that all derivatives of order  n>2  will be  0  for  x2 .

The nth derivative of  sin x  is given by  sin(x+nπ2) . So,

dndxnu×v=sin(x+nπ2)×x2+n×sin(x+(n−1)π2)×2x+n(n−1)2×sin(x+(n−2)π2)×2