Simplify $\frac1{\sqrt{x^2+1}}-\frac{x^2}{(x^2+1)^{3/2}}$

Question

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.

Answer 1

$\sqrt x$ is just a shorthand for $x^{1/2}$. Hence we can multiply the two halves of the first fraction in the first term by $x^2+1$: $$\frac1{\sqrt{x^2+1}}-\frac{x^2}{(x^2+1)^{3/2}}=\frac{x^2+1}{(x^2+1)^{3/2}}-\frac{x^2}{(x^2+1)^{3/2}}$$ and the target expression follows.

Answer 2

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

Answer 3

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

Answer 4

Factor from $\dfrac{1}{\sqrt{x^2+1}}$. You will have:

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