What is the derivative of this function ${csc}^{- 1} (3 x)$ $csc^-1(3x)$ ?

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Answer 1 · 2017-07-04T22:35:49

$\frac{d}{d x} ({csc}^{- 1} (3 x)) = - \frac{1}{(\sqrt{9 - \frac{1}{x^{2}}}) x^{2}}$ $d/(dx) (csc^-1(3x)) = color(blue)(-1/((sqrt(9-1/(x^2)))x^2)$

We can use the chain rule...

$\frac{d}{d x} ({csc}^{- 1} (3 x)) = \frac{d {csc}^{- 1} (u)}{d u} \frac{d u}{d x}$ $d/(dx) (csc^-1(3x)) = (dcsc^-1(u))/(du) (du)/(dx)$

where

$u = 3 x$ $u = 3x$

and

$\frac{d}{d u} ({csc}^{- 1} (u)) = - \frac{1}{(\sqrt{1 - \frac{1}{u^{2}}}) u^{2}}$ $d/(du) (csc^-1(u)) = -1/((sqrt(1-1/(u^2)))u^2)$ :

$= - \frac{\frac{d}{d x} (3 x)}{(9 \sqrt{1 - \frac{1}{9 x^{2}}}) x^{2}}$ $= -(d/(dx)(3x))/((9sqrt(1-1/(9x^2)))x^2)$

Factor out the constant, $3$ $3$ :

$= - \frac{3 \frac{d}{d x} (x)}{(9 \sqrt{1 - \frac{1}{9 x^{2}}}) x^{2}}$ $= -(3d/(dx)(x))/((9sqrt(1-1/(9x^2)))x^2)$

Simplify the $\frac{3}{9}$ $3/9$ quantity:

$= - \frac{\frac{d}{d x} (x)}{(3 \sqrt{1 - \frac{1}{9 x^{2}}}) x^{2}}$ $= -(d/(dx)(x))/((3sqrt(1-1/(9x^2)))x^2)$

And the derivative of $x$ $x$ is $1$ $1$ , according to the power rule:

$= - \frac{1}{(3 \sqrt{1 - \frac{1}{9 x^{2}}}) x^{2}}$ $= color(blue)(-1/((3sqrt(1-1/(9x^2)))x^2)$

or

$color(blue)(-1/((sqrt(9-1/(x^2)))x^2)$

What is the derivative of this function csc^-1(3x)csc−1(3x)csc^-1(3x)?