y = tan^-1(sqrt(5x^2-1)) find dy/dx

Answer 1

just use the chain rule

1/(1+(√(5x^2-1))^2) * 5x/√(5x^2-1)
= 5x / √(5x^2-1)(1+5x^2-1)
= 1/x√(5x^2-1)

You can also do it by implicit differentiation

tan y = √(5x^2-1)
sec^2 y y' = 5x/√(5x^2-1)

since tan y = √(5x^2-1), sec^2y = 1+tan^2 = 5x^2

y' = 5x/(5x^2√(5x^2-1))
= 1/x√(5x^2-1)