Example 1: y = tan^(-1)((sqrt(1+x^2) - 1)/x) Substitute x = tan t: sqrt(1+tan^2 t) = sec t. Argument = (sec t - 1)/tan t = (1 - cos t)/(sin t) = tan(t/2). So y = tan^(-1)(tan(t/2)) = t/2 = (1/2)*tan^(-1)(x). dy/dx = 1/(2(1+x^2)).

Example 2: y = sin^(-1)((2^(x+1))/(1+4^x)) Let 2^x = t. Then 2^(x+1) = 2t and 4^x = t^2. y = sin^(-1)(2t/(1+t^2)). Put t = tan(u): 2t/(1+t^2) = sin(2u). y = sin^(-1)(sin(2u)) = 2u = 2*tan^(-1)(2^x) [if |2u| <= pi/2]. dy/dx = 2 * 1/(1+4^x) * 2^x * ln 2 = (2^(x+1) * ln 2)/(1+4^x).

Example 3: y = cos^(-1)((1-x^2)/(1+x^2)) Put x = tan t: (1-tan^2 t)/(1+tan^2 t) = cos(2t). y = cos^(-1)(cos(2t)) = 2t = 2*tan^(-1)(x) [for x >= 0]. dy/dx = 2/(1+x^2).

The pattern: inverse trig of a recognizable expression -> simplify -> trivial derivative.