Simplify $\frac1{\sqrt{x^2+1}}-\frac{x^2}{(x^2+1)^{3/2}}$ I want to know why
$$\frac1{\sqrt{x^2+1}} - \frac{x^2}{(x^2+1)^{3/2}}$$ 
can be simplified into
$$\frac1{(x^2+1)^{3/2}}$$
I tried to simplify by rewriting radicals and fractions. I was hoping to see a clever trick (e.g. adding a clever zero, multiplying by a clever one? Quadratic completion?)  
\begin{align}
\frac{1}{\sqrt{x^2+1}} - \frac{x^2}{(x^2+1)^{3/2}} & = \\
& = (x^2+1)^{-1/2} -x^2*(x^2+1)^{-3/2} \\
& = (x^2+1)^{-1/2} * ( 1 - x^2 *(x^2+1)^{-1}) \\
& = ...
\end{align}
To give a bit more context, I was calculating the derivative of $\frac{x}{\sqrt{x^2+1}}$ in order to use newtons method for approximating the roots.

HINT: we have $$\frac{1}{\sqrt{x^2+1}}-\frac{x^2}{\sqrt{x^2+1}^3}=\frac{x^2+1-x^2}{\sqrt{x^2+1}^3}$$
