di65fferenciate with respect to x : y= log { ( √ x+1 + √ x-1) / ( √x+1 - √x-1) }

Question

Please give step by step answer.

Answer 1

Assume log≡ln.

y=ln{(√(x+1)+√(x-1))/(√(x+1)-√(x-1))}

First rationalise the argument by multiplying top and bottom by √(x+1)+√(x-1)

(√(x+1)+√(x-1))²/((x+1)-(x-1))=(x+1+2√(x²-1)+x-1)/2=x+√(x²-1).

y=ln(x+√(x²-1)),

dy/dx=(1/(x+√(x²-1))(1+x/√(x²-1))=

(1/(x+√(x²-1))(x+√(x²-1))/√(x²-1))=1/√(x²-1)).