Question

differentiate f(x)=(sqroot (x^2+1))(sqroot(6x+5)) everything divided by (sqroot(x^2-1))

Answer 1

combine them all into the one square root:

sqrt[(x^2 + 1)(6x + 5)/(x^2 - 1)]

so if we call whatever's inside the square root g, then by the chain
rule, the derivative is g'/2sqrt(g)

so to find g', you gotta use the product rule and the quotient rule:

g' = [[2x(6x + 5) + 6(x^2 + 1)](x^2 - 1) - 2x(x^2 + 1)(6x + 5)]/(x^2 - 1)^2


= 2(3x^4 - 12x^2 - 10x - 3)/(x^2 - 1)^2

so the derivative overall is:

(3x^4 - 12x^2 - 10x - 3)/sqrt(g)*(x^2 - 1)^2

= (3x^4 - 12x^2 - 10x - 3)*sqrt(x^2 - 1)/[sqrt(x^2 + 1)*sqrt(6x + 5)*(x^2 - 1)^2]

Answer 2

guess it can be simplified to:

(3x^4 - 12x^2 - 10x - 3)/[sqrt(x^4 - 1)*sqrt(6x + 5)*(x^2 - 1)]