" (2) "li1*1+{(x+(x^(2))/(1+x^(2)));x(-R)" be "

The derivative of sin^(-1) ((2x)/(1+x^(2))) w.r.t sin^(-1) ((1-x^(2))/(1+x^(2))) is

The derivative of Cos ^(-1) "" (1- x ^(2))/( 1 + x ^(2)) w.r.t. Tan ^(-1) "" (2x)/( 1 - x ^(2)) is

The derivative of Tan ^(-1)"" (sqrt(1 + x ^(2))-1)/(x) w.r.t. Tan ^(-1) "" (2x sqrt(1-x ^(2)))/(1 - 2 x ^(2))at x =0 is

Let f : R to and f (x) = (x (x^(4) + 1) (x+1) +x ^(4)+2)/(x^(2) +x+1), then f (x) is :

Let f : R to and f (x) = (x (x^(4) + 1) (x+1) +x ^(4)+2)/(x^(2) +x+1), then f (x) is :

Let (x, y) be such that sin^(-1)(ax)+cos^(-1)(y)+cos^(-1)(bxy)=pi//2 . Match the statements in column I with statements in column II. {:("Column I", "Column II"), ("A) If a = 1 and b = 0, then (x, y)", "p) lies on the circle "x^(2)+y^(2)=1), ("B) If a = 1 and b = 1, then (x, y)", "q) lies on "(x^(2)-1)(y^(2)-1)=0), ("C) If a = 1 and b = 2, then (x, y)", "r) lies on y = x"), ("D) If a = 2 and b = 2, then (x, y)", "s) lies on "(4x^(2)-1)(y^(2)-1)=0):}

If P_(r) is the coefficient of x^® in the expansion of (1 + x)^(2) (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)... prove that P_(r) = (2^(2))/((2^(r) -1))(P_(r-1) + P_(r-2) )and P_(4) = (1072)/(315) .