derivatives - Why is my answer incorrect for this differentiation question?

$$y = x* ((x^2+1)^{1/2})$$
I must find $$dy/dx$$

$$u = x, v = (x^2+1)^{1/2}$$
To do this I must use the product rule and the chain rule.
To get dv/dx,
$$(dv/dx) = (1/2)*(b)^{-1/2}*2x $$

$$(dv/dx) = x*(b)^{-1/2} $$
$$(dv/dx) = x*(x^2+1)^{-1/2} $$
$$(dv/dx) = x*\frac{1}{\sqrt{x^2+1}} $$
$$(dv/dx) = \frac{x}{\sqrt{x^2+1}} $$
so now $$u*(dv/dx) = x*\frac{x}{\sqrt{x^2+1}} $$
$$u*(dv/dx) = \frac{x^2}{\sqrt{x^2+1}} $$
now for v* du/dx

$$v*du/dx = 1 * ((x^2+1)^{1/2}) =(x^2+1)^{1/2} $$

so adding the parts together as follows :
$$(u*(dv/dx))+(v*(du/dx))$$

gives:
$$\frac{x^2}{\sqrt{x^2+1}} +(x^2+1)^{1/2} $$

which could be shown as

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

so far I feel confident with my workings out, here is what I do next.

$$\frac{(2x^4 +x^2)^{1/2}}{\sqrt{x^2+1}}$$

as I have multiplied the numerators together.
This answer is incorrect as I am aware the correct answer is :
$$\frac{2x^2+1}{(x^2+1)^{1/2}}$$

Can someone please show me where I have gone wrong? Also show the correct course of action to solve this ? I realise the denominator is the same. Any help is very much appreciated. Thanks

Answer

As Mattos said, the problem uses $x^{2} + 1$ and you used $x^{2} - 1$. Otherwise you were ok to the point where you said you were confident. At that point you have two fractions to add. You need to get a common denominator and add the numerators, not multiply the numerators.