Simplify $\frac{1}{\sqrt{{x}^{2}+1}}-\frac{{x}^{2}}{({x}^{2}+1{)}^{3/2}}$

I want to know why

$$\frac{1}{\sqrt{{x}^{2}+1}}-\frac{{x}^{2}}{({x}^{2}+1{)}^{3/2}}$$

can be simplified into

$$\frac{1}{({x}^{2}+1{)}^{3/2}}$$

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?)

$$\begin{array}{rl}\frac{1}{\sqrt{{x}^{2}+1}}-\frac{{x}^{2}}{({x}^{2}+1{)}^{3/2}}& =\\ & =({x}^{2}+1{)}^{-1/2}-{x}^{2}\ast ({x}^{2}+1{)}^{-3/2}\\ & =({x}^{2}+1{)}^{-1/2}\ast (1-{x}^{2}\ast ({x}^{2}+1{)}^{-1})\\ & =...\end{array}$$

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.

I want to know why

$$\frac{1}{\sqrt{{x}^{2}+1}}-\frac{{x}^{2}}{({x}^{2}+1{)}^{3/2}}$$

can be simplified into

$$\frac{1}{({x}^{2}+1{)}^{3/2}}$$

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?)

$$\begin{array}{rl}\frac{1}{\sqrt{{x}^{2}+1}}-\frac{{x}^{2}}{({x}^{2}+1{)}^{3/2}}& =\\ & =({x}^{2}+1{)}^{-1/2}-{x}^{2}\ast ({x}^{2}+1{)}^{-3/2}\\ & =({x}^{2}+1{)}^{-1/2}\ast (1-{x}^{2}\ast ({x}^{2}+1{)}^{-1})\\ & =...\end{array}$$

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.