Calculus problems: Finding the derivative...HELP!?

Question

Well, these are only two problems left. HELP!!

Find the derivative of the ff:
1.) y = x/√x-1
(y is equal to x over square root of x-1)

2.) y = √x+1/x
(y is equal to square root of x+1 over x)


Please HELP...this is for tomorrow!

Thanks in advance.

The x shall be substituted by x+∆x.

Answer 1

Question 1
y = x / (x - 1)^(1/2)
dy/dx is given by:-
[ (x - 1)^(1/2) - x.(1/2).(x - 1)^(-1/2) ] / (x - 1)
= (x - 1)^(-1/2).[ x - 1 - x/2 ] / (x - 1)
= (x/2 - 1) / (x - 1)^(1/2)
= (x - 2) / [ 2.(x - 1)^(1/2) ]

Question 2
dy/dx is given by:-
[ x.(1/2).(x + 1)^(-1/2) - (x + 1)^(1/2) ] / x²
= (x + 1)^(-1/2).[ 2x -2(x+ 1) ] / (2x²)
= 1 / [ (x + 1)^(1/2).2x² ]

Answer 2

1.) y = x/√x-1
dy/dx= [√(x-1)*1-x*1/2(x-1)^-1/2]/ (√x-1)^2
dy/dx= [√(x-1) -x/(2√x-1)]/ (x-1)
etc

2.) y = √x+1/x
dy/dx=1/2*x^-1/2 + (-1)x^-2
dy/dx = 1/(2√x) - 1 /x^2

Answer 3

1.
y= x * (x-1)^(-1/2)
y' = 1*(x-1)^(-1/2) + x*(-1/2)*(x-1)^(-3/2)
= 1/(sqroot(x-1)) - x/(2* (sqroot(x-1)^3)

2.
y=x^(1/2) + x^(-1)
y' = 1/2 * 1/sqroot(x) - 1/x^2

hope this help