What is the derivative of ${(arctan x)}^{3}$ $(arctan x)^3$ ?

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What is the derivative of ${(arctan x)}^{3}$ $(arctan x)^3$ ?

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Answer 1 · 2017-04-11T09:23:12

$\frac{d}{d x} {arctan}^{3} x = \frac{3 {arctan}^{2} x}{x^{2} + 1}$ $d/dx arctan^3x = (3arctan^2x) /(x^2+1)$

Explanation:

By the chain rule we have:

$\frac{d}{d x} {(arctan x)}^{3} = \frac{d}{d x} {arctan}^{3} x$ $d/dx (arctanx)^3 = d/dx arctan^3x$
$= 3 {arctan}^{2} x \cdot \frac{d}{d x} (arctan x)$ $" " = 3arctan^2x* d/dx (arctanx)$

And:

$\frac{d}{d x} (arctan x) = \frac{1}{x^{2} + 1}$ $d/dx(arctanx) = 1/(x^2+1)$

And so:

$\frac{d}{d x} {arctan}^{3} x = \frac{3 {arctan}^{2} x}{x^{2} + 1}$ $d/dx arctan^3x = (3arctan^2x) /(x^2+1)$

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What is the derivative of ${(arctan x)}^{3}$ $(arctan x)^3$ ?

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