Derivative of $$$\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}$$$

The function $$$\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \operatorname{asin}{\left(u \right)}$$$ and $$$g{\left(x \right)} = \frac{2 x}{x^{2} + 1}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\operatorname{asin}{\left(u \right)}\right) \frac{d}{dx} \left(\frac{2 x}{x^{2} + 1}\right)\right)}$$

The derivative of the inverse sine is $$$\frac{d}{du} \left(\operatorname{asin}{\left(u \right)}\right) = \frac{1}{\sqrt{1 - u^{2}}}$$$:

$${\color{red}\left(\frac{d}{du} \left(\operatorname{asin}{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{2 x}{x^{2} + 1}\right) = {\color{red}\left(\frac{1}{\sqrt{1 - u^{2}}}\right)} \frac{d}{dx} \left(\frac{2 x}{x^{2} + 1}\right)$$

Return to the old variable:

$$\frac{\frac{d}{dx} \left(\frac{2 x}{x^{2} + 1}\right)}{\sqrt{1 - {\color{red}\left(u\right)}^{2}}} = \frac{\frac{d}{dx} \left(\frac{2 x}{x^{2} + 1}\right)}{\sqrt{1 - {\color{red}\left(\frac{2 x}{x^{2} + 1}\right)}^{2}}}$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = \frac{x}{x^{2} + 1}$$$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(\frac{2 x}{x^{2} + 1}\right)\right)}}{\sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = \frac{{\color{red}\left(2 \frac{d}{dx} \left(\frac{x}{x^{2} + 1}\right)\right)}}{\sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}}$$

Apply the quotient rule $$$\frac{d}{dx} \left(\frac{f{\left(x \right)}}{g{\left(x \right)}}\right) = \frac{\frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} - f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)}{g^{2}{\left(x \right)}}$$$ with $$$f{\left(x \right)} = x$$$ and $$$g{\left(x \right)} = x^{2} + 1$$$:

$$\frac{2 {\color{red}\left(\frac{d}{dx} \left(\frac{x}{x^{2} + 1}\right)\right)}}{\sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = \frac{2 {\color{red}\left(\frac{\frac{d}{dx} \left(x\right) \left(x^{2} + 1\right) - x \frac{d}{dx} \left(x^{2} + 1\right)}{\left(x^{2} + 1\right)^{2}}\right)}}{\sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}}$$

The derivative of a sum/difference is the sum/difference of derivatives:

$$\frac{2 \left(- x {\color{red}\left(\frac{d}{dx} \left(x^{2} + 1\right)\right)} + \left(x^{2} + 1\right) \frac{d}{dx} \left(x\right)\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = \frac{2 \left(- x {\color{red}\left(\frac{d}{dx} \left(x^{2}\right) + \frac{d}{dx} \left(1\right)\right)} + \left(x^{2} + 1\right) \frac{d}{dx} \left(x\right)\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}}$$

The derivative of a constant is $$$0$$$:

$$\frac{2 \left(- x \left({\color{red}\left(\frac{d}{dx} \left(1\right)\right)} + \frac{d}{dx} \left(x^{2}\right)\right) + \left(x^{2} + 1\right) \frac{d}{dx} \left(x\right)\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = \frac{2 \left(- x \left({\color{red}\left(0\right)} + \frac{d}{dx} \left(x^{2}\right)\right) + \left(x^{2} + 1\right) \frac{d}{dx} \left(x\right)\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:

$$\frac{2 \left(- x {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \left(x^{2} + 1\right) \frac{d}{dx} \left(x\right)\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = \frac{2 \left(- x {\color{red}\left(2 x\right)} + \left(x^{2} + 1\right) \frac{d}{dx} \left(x\right)\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$\frac{2 \left(- 2 x^{2} + \left(x^{2} + 1\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = \frac{2 \left(- 2 x^{2} + \left(x^{2} + 1\right) {\color{red}\left(1\right)}\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}}$$

Simplify:

$$\frac{2 \left(1 - x^{2}\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = - \frac{2 \left(x - 1\right) \left(x + 1\right)}{\left(x^{2} + 1\right) \left|{x - 1}\right| \left|{x + 1}\right|}$$

Thus, $$$\frac{d}{dx} \left(\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}\right) = - \frac{2 \left(x - 1\right) \left(x + 1\right)}{\left(x^{2} + 1\right) \left|{x - 1}\right| \left|{x + 1}\right|}.$$$