If `y= (ax^(2))/((x-a)(x-b) (x-c)) + (bx)/((x-b) (x-c))+ (c )/((x-c)) + 1` then prove that `(y')/(y)= (1)/(x) [(a)/(a-x) + (b)/(b-x) + (c)/(c-x)]`

If f(x)= y = (ax-b)/(cx-a) , then prove that f(y) = x

If g(x)=(f(x))/((x-a)(x-b)(x-c)),w h e r ef(x) is a polynomial of degree < 3, then prove that (dg(x))/(dx)=|{:(1,a, f(a)(x-a)^(-2)),(1,b,f(b)(x-b)^(-2)), (1,c,f(c)(x-c)^(-2)):}|/|{:(a^2,a,1),(b^2,b,1),(c^2,c,1):}|

If x-iy = sqrt((a+ib)/(c-id)) , prove that (x^(2) + y^(2))^(2) = (a^(2) + b^(2))/(c^(2) + d^(2))

If y= |(f(x),g(x),h(x)),(l,m,n),(a,b,c)| , prove that (dy)/(dx)= |(f'(x),g'(x),h'(x)),(l,m,n),(a,b,c)|

If a, b, x, y are positive natural numbers such that (1)/(x) + (1)/(y) = 1 then prove that (a^(x))/(x) + (b^(y))/(y) ge ab .

Show that ((x+b)(x+c))/((b-a)(c-a))+((x+c)(x+a))/((c-b)(a-b))+((x+a)(x+b))/((a-c)(b-c))=1 is an identity.

If f(x)=|[(x-a)^4, (x-a)^3, 1] , [(x-b)^4, (x-b)^3,1] , [(x-c)^4, (x-c)^3,1]| then f'(x)=lambda|[(x-a)^4,(x-a)^2,1] , [(x-b)^4, (x-b)^2, 1] , [(x-c)^4, (x-c)^2,1]| . Find the value of lambda

If x, y in R and |{:((a ^(x) + a ^(-x) ) ^(2) , (a ^(x) - a ^(-x) ) ^(2), 1),((b ^(x) + b ^(-x)) ^(2),(b ^(x) - b ^(-x) ) ^(2) , 1), ( ( c ^(x) + c ^(-x)) ^(2) , (c ^(x) - c ^(-x)) ^(2) , 1):}|=2 y + 6 then y = ____________.

If the points A(x,y), B(a,0) and C(0,b) are collinear, prove that (x)/(a) + (y)/(b) = 1

If x=1+(log)_a b c , y=1+(log)_b c a and z=1+(log)_c a b , then prove that x y z=x y+y z+z x