Consider the normal elliptic integral of first and second kind (completes), for $k \in (0,1)$ $$K(k)= \int^{\frac{\pi}{2}}_{0} \frac{d\theta}{\sqrt{1-k^2\sin \theta}}$$ and $$E(k)=\int^{\frac{\pi}{2}}_{0} {\sqrt{1-k^2\sin \theta}} \; d\theta.$$
Now consider the function $f:(0,1) \longrightarrow \mathbb{R}$ given by $$f(k)=K(k)E(k)-k'^2K^2(k), \; \forall \; k \in (0,1),$$ where $k'^2=1-k^2$.
I want to prove that $f$ differentiable strictly increasing. Of course $f$ is differentiable. First of all, is that true? If so, how do I prove it?