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Step-by-step Solution

Derive the function arcsin(((1-1*x^2)/(1+x^2))) with respect to x

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Answer

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

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(arcsin\left(\frac{1-x^2}{1+x^2}\right)\right)$
1

Taking the derivative of arcsine

$\frac{1}{\sqrt{1-\left(\frac{1-x^2}{1+x^2}\right)^2}}\cdot\frac{d}{dx}\left(\frac{1-x^2}{1+x^2}\right)$
2

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{1}{\sqrt{1-\left(\frac{1-x^2}{1+x^2}\right)^2}}\cdot\frac{\left(1+x^2\right)\frac{d}{dx}\left(1-x^2\right)-\left(1-x^2\right)\frac{d}{dx}\left(1+x^2\right)}{\left(1+x^2\right)^2}$

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Answer

$\frac{1}{\sqrt{1-\left(\frac{1-x^2}{1+x^2}\right)^2}}\cdot\frac{-2x\left(1+x^2\right)-2x\left(1-x^2\right)}{\left(1+x^2\right)^2}$
$\frac{d}{dx}\left(arcsin\left(\frac{1-x^2}{1+x^2}\right)\right)$

Main topic:

Polynomials

Time to solve it:

~ 1.95 seconds