Differentiating $\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ 
Find the derivative with respect to $x$ of 
  $$\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$$

The partial solution to this problem is given as follows:
$y=\left(\frac{x}{\sqrt{1-x^2}}\right)$
Then:
$\frac{dy}{dx} = (1-x^2)^{-3/2}$ 

Can you show the steps in arriving at this partial solution of $\frac{dy}{dx}$?

Product Rule with Chain Rule, Quotient Rule with Chain Rule, and I end up with additional terms.  Notably, an $x^2$ multiplying the $(1-x^2)^{-3/2}$, plus an additional term of $(1-x^2)^{-1/2}$ 

Continuing to work through the solution provided I have arrived at an answer.  It is as follows:
$\frac{d}{dx} \frac{x}{\sqrt(1-x^2)}$
$\frac{d}{dx} x(1-x^2)^{-1/2}$
Using the Product Rule (u' * v) + (v' * u) where u = $x$ and v = $(1-x^2)^{-1/2}$ and combining with the Chain Rule for the derivation of v' yields the following:
$(1)*(1-x^2)^{-1/2} + (\frac{-1}{2}(1-x^2)^{-3/2}*(-2x)*(x)$
NOTE: The Chain Rule applied to v' gave us this: $\frac{-1}{2}(1-x^2)^{-3/2}*(-2x)$
Now, further simplifying ...
$(1-x^2)^{-1/2} + (\frac{2x^2}{2}(1-x^2)^{-3/2}$
$(1-x^2)^{-1/2} + (x^2(1-x^2)^{-3/2}$
The above was where I was stuck, but here is the breakthrough ...
$(1-x^2)^{-1/2} + \left(\frac{x^2}{(1-x^2)^{3/2}}\right)$
Get a common denominator ... 
$\frac{(1-x^2)^{-1/2}(1-x^2)^{3/2} + x^2}{(1-x^2)^{3/2}}$
$\frac{(1-x^2)^{2/2} + x^2}{(1-x^2)^{3/2}}$
$\frac{1-x^2+x^2}{(1-x^2)^{3/2}}$
$\frac{1}{(1-x^2)^{3/2}}$
$=(1-x^2)^{-3/2}$
NOTE:  This is only the partial solution to the problem.  For those interested in the full solution read on ...
Without showing this derivation, I know 
$\frac{d}{dx} tan^{-1}(x) = \frac{1}{1+x^2}$
We also know from the original problem that $x = \frac{x}{\sqrt(1-x^2)}$
So now substitute $x = \frac{x}{\sqrt(1-x^2)}$ into derivation of $\frac{d}{dx} tan^{-1}(x) = \frac{1}{1+x^2}$
This yields 
$\frac{d}{dx} tan^{-1}(x) = \frac{1}{1+x^2} = \frac{1}{1+\left(\frac{x}{\sqrt(1-x^2)} \right)^2}$
Square the term in the denominator
$=\frac{1}{1+\frac{x^2}{1-x^2}}$
Get a common denominator
$=\frac{1}{\frac{1-x^2+x^2}{1-x^2}}$
Simplify
$=1-x^2$
Now, finishing up ... 
The derivative with respect to $x$ of $tan^-1\left(\frac{x}{\sqrt(1-x^2)}\right)$ requires the Chain Rule.  This is:
$\frac{d}{dx}tan^{-1}\left(\frac{x}{\sqrt(1-x^2)}\right) * \frac{d}{dx}\frac{x}{\sqrt(1-x^2)}$
So, the derivative of the outside function $\frac{d}{dx}tan^{-1}\left(\frac{x}{\sqrt(1-x^2)}\right)$, or think of it just like the derivative of the $tan^{-1}(x)$ where we later substitute back in $x = \frac{x}{\sqrt(1-x^2)}$, times the derivative of the inside function $\frac{d}{dx}\frac{x}{\sqrt(1-x^2)}$.  We have already found both derivations above.  Let's substitute the derivations in and finish solving this problem.
By Chain Rule:
$\frac{d}{dx}tan^{-1}\left(\frac{x}{\sqrt(1-x^2)}\right) = \frac{d}{dx}\left(tan^{-1}\left(\frac{x}{\sqrt(1-x^2)}\right)\right) * \frac{d}{dx}\left(\frac{x}{\sqrt(1-x^2)}\right)$
$=(1-x^2) * (1-x^2)^{-3/2}$
$=(1-x^2)^{2/2} * (1-x^2)^{-3/2}$
$=(1-x^2)^{-1/2}$
$=\frac{1}{\sqrt(1-x^2)}$
