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

The derivative of arctan(x/(1+x)) is 1/(2x^2+2x+1)

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

The first derivative of arctan(x/(1+x)) is 1/(2x^2+2x+1)

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

\frac{1}{2 x ^{2} + 2 x + 1}

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{1}{2 x ^{2} + 2 x + 1}

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