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

Simplify

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