Calculus of a Single Variable, Chapter 5, 5.8, Section 5.8, Problem 73

The expression of this function includes difference, product and two table functions (except the polynomials), sinh^(-1)(z) and sqrt(z).
The difference rule is (u-v)' = u' - v', the product rule is (uv)' = u'v + uv', the chain rule is (u(v(x)))' = u'(v(x))*v'(x).
The derivative of sinh^(-1)(z) is 1/sqrt(1 + z^2), the derivative of sqrt(z) is 1/(2sqrt(z)).
These rules together give us
y' = 2sinh^-1(2x) + 2x ((2x)')/sqrt(1+(2x)^2) - ((4x^2)')/(2sqrt(1+(2x)^2)) =
= 2sinh^-1(2x) + 2x (2)/sqrt(1+4x^2) - (8x)/(2sqrt(1+4x^2)) =
= 2sinh^-1(2x).