[" Ex.9.If "x+y=1," prove that "],[qquad (d^(n))/(dx^(n))(x^(n)y^(n))=n![y^(n)-(^(n)C_(1))^(2)y^(n-1)x+(^(n)C_(2))^(2)y^(n-2)x^(2)+...+(-1)^(n)x^(n)]]

If x + y = 1, show that D ^ (n) (x ^ (n) y ^ (n)) = n! [Y ^ (n) - (nC_ (1)) ^ (2) y ^ (n -1) x + (nC_ (2)) ^ (2) y ^ (n-2) x ^ (2) + ...... + (- 1) ^ (n) x ^ (n)]

(x^(2^(n-1))+y^(2^(n-1)))(x^(2^(n-1))-y^(2^(n-1)))=

(x^(2^n)-y^(2^n))/(x^(2^(n-1))+y^(2^(n-1)))=

(d^(n))/(dx^(n))(log x)=(a)((n-1)!)/(x^(n))(b)(n!)/(x^(n))(c)((n-2)!)/(x^(n))(d)(-1)^(n-1)((n-1)!)/(x^(n))

Show : x-^(n)C_(1)(x+y)+^(n)C_(2)(x+2y)-^(n)C_(3)(x+3y)+....=0

If y=((1+x)/(1-x))^(n) , prove that (1-x^(2))(d^(2)y)/(dx^(2))=2(n+x)(dy)/(dx).

Simplify : (x^(2^(n-1)) + y^(2^(n-1)))(x^(2^(n-1)) - y^(2^(n-1)))

If y=x^(n-1)logx , Prove that, x^2(d^2y)/(dx^2)+(3-2n)xdy/dx+(n-1)^2y=0