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

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Answer 1 · 2017-07-04T22:35:49

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

We can use the chain rule...

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

where

$u = 3x$

and

$d/(du) (csc^-1(u)) = -1/((sqrt(1-1/(u^2)))u^2)$ :

$= -(d/(dx)(3x))/((9sqrt(1-1/(9x^2)))x^2)$

Factor out the constant, $3$ :

$= -(3d/(dx)(x))/((9sqrt(1-1/(9x^2)))x^2)$

Simplify the $3/9$ quantity:

$= -(d/(dx)(x))/((3sqrt(1-1/(9x^2)))x^2)$

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

$= color(blue)(-1/((3sqrt(1-1/(9x^2)))x^2)$

or

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

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