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

##### 1 Answer
Jul 4, 2017

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

#### Explanation:

We can use the chain rule...

$\frac{d}{\mathrm{dx}} \left({\csc}^{-} 1 \left(3 x\right)\right) = \frac{\mathrm{dc} s {c}^{-} 1 \left(u\right)}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$

where

$u = 3 x$

and

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

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

Factor out the constant, $3$:

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

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

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

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)