Why is Implicit Differentiation needed for Derivative of $y = \arcsin (2x+1)$? my function is: $y = \arcsin (2x+1)$ and I want to find its derivative.
My approach was to apply the chain rule: 
$$y' = \frac{dg}{du} \frac{du}{dx}$$
with $g = \arcsin(u)$ and $u = 2x+1$. 
$$g' = \frac{1}{\sqrt{1-u^2}}.$$
${u}' = 2$. 
My solution therefore was
$$\frac{1}{\sqrt{1-u^2}} \cdot 2 = \frac{2}{\sqrt{1-(2x+1)^2}}.$$
This seems to be wrong and the correct solution is given by: 
$\frac{1}{\sqrt{-x^{2}-x}}$
I know that implicit differentiation should be used for this particular problem, but I do not really understand why. I appreciate any help!

Your solution is correct; they just simplified it further:
\begin{align*}
\frac{2}{\sqrt{1 - (2x + 1)^2}}
&= \frac{2}{\sqrt{1 - (4x^2 + 4x + 1)}} \\
&= \frac{2}{\sqrt{-4x^2 - 4x}} \\
&= \frac{2}{\sqrt{4(-x^2 - x)}} \\
&= \frac{2}{\sqrt{4}\sqrt{-x^2 - x}} \\
&= \frac{2}{2\sqrt{-x^2 - x}} \\
&= \frac{1}{\sqrt{-x^2 - x}} \\
\end{align*}
