What is the derivative of arcsin(sqrt(x/(1-x))) ?

The derivative of arcsin(sqrt(x/(1-x))) is 1/(2sqrt(x)(-x+1)\sqrt{-2x+1)}

What is the first derivative of arcsin(sqrt(x/(1-x))) ?

The first derivative of arcsin(sqrt(x/(1-x))) is 1/(2sqrt(x)(-x+1)\sqrt{-2x+1)}

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\frac{d}{d x} (arcsin (\frac{x}{1 - x}))

\frac{1}{2 x ( - x + 1 ) - 2 x + 1}

Solution steps

\frac{d}{d x} (arcsin (\frac{x}{1 - x}))

Apply the chain rule:

\frac{1}{1 - ( \frac{x}{1 - x} ) ^{2}} \frac{d}{d x} (\frac{x}{1 - x})

= \frac{1}{1 - ( \frac{x}{1 - x} ) ^{2}} \frac{d}{d x} (\frac{x}{1 - x})

\frac{d}{d x} (\frac{x}{1 - x}) = \frac{1}{2 x ( - x + 1 ) ^{\frac{3}{2}}}

= \frac{1}{1 - ( \frac{x}{1 - x} ) ^{2}} \cdot \frac{1}{2 x ( - x + 1 ) ^{\frac{3}{2}}}

Simplify

\frac{1}{1 - ( \frac{x}{1 - x} ) ^{2}} \cdot \frac{1}{2 x ( - x + 1 ) ^{\frac{3}{2}}} : \frac{1}{2 x ( - x + 1 ) - 2 x + 1}

= \frac{1}{2 x ( - x + 1 ) - 2 x + 1}

\int \frac{x}{x ^{2}} d x

\int \frac{sin ( x + 1 )}{x + 1} d x

\int_{1}^{e} \frac{ln ( x )}{x} d x

(1 + 6 x^{2}) (x - x^{2})

x \to \infty lim (- 3 x^{2})

What is the derivative of arcsin(sqrt(x/(1-x))) ?
The derivative of arcsin(sqrt(x/(1-x))) is 1/(2sqrt(x)(-x+1)\sqrt{-2x+1)}
What is the first derivative of arcsin(sqrt(x/(1-x))) ?
The first derivative of arcsin(sqrt(x/(1-x))) is 1/(2sqrt(x)(-x+1)\sqrt{-2x+1)}