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

The derivative of arctan(sqrt((1+x/(1-x)))) is 1/(2sqrt(x+1)(-x+1)^{1/2)}

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

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

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\frac{d}{d x} (arctan (\frac{1 + x}{1 - x}))

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

Solution steps

\frac{d}{d x} (arctan (\frac{1 + x}{1 - x}))

Apply the chain rule:

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

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

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

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

Simplify

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

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

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