`(d^(20))/(dx^(20))(2cosxcos3x)` is equal to........

(d^(20))/(dx^(20))(2cos x cos3x) is equal to (a) 2^(20)(cos2x-2^(20)cos3x)(b)2^(20)(cos2x+2^(20)cos4x)(c)2^(20)(cos2x+2^(20)sin4x)(d)2^(20)(sin2x-2^(20)sin4x)

sum_(k=0)^(20)(.^(20)C_(k))^(2) is equal to

(d^(20))/(dx^(20))(2cos x cos3x)=2^(20)(cos2x-2^(20)cos4x)(b)2^(20)(cos2x+2^(20)cos4x)(c)2^(20)(sin2x+2^(20)sin4x)(d)2^(20)(sin2x-2^(20)sin4x)

Match the following : {:("Column I ", " Column II" ), ("(A)" if lim_(x to 1) (1-x) tan""(pix)/2 = k " then " sin (1/k) " is" , "(p)"4),( "(B)" if lim_(x to 5) (x^(k)-5^(k))/(x-5) = 500 " then k is " , "(q)" 1),("(C)" lim_(x to oo)(1 + 4/(x+1))^((3x-1)/3) " is equal to " e^(k) " , then k is" , "(r) A perfect sqare"), ("(D) " d^(20)/(dx^(20)) (2 cos x"," cos 3x)= 2^(4k) [ cos2x + 2^(20) . cos4k]" then k is " , "(s)" 5),(, "(t)An odd number"):}

If y^(2)=P(x) is polynomial of degree 3, then 2((d)/(dx))(y^(3)*d^(2)(y)/(dx^(2))) is equal to

d/(dx) (e^(x^3)) is equal to

If (d^(2)x)/(dy^(2)) ((dy)/(dx))^(3) + (d^(2) y)/(dx^(2)) = k , then k is equal to