Let 2^x = tanθ

Considering the limits,

–∞ < x < 0

2^–∞ < 2^x < 2⁰

0 < tanθ < 1

Now,

y = tan^–1(tan2θ)

y = 2θ

y = 2tan^–1(2^x)

Differentiating w.r.t x, we get