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)#