# What is the derivative of y = xsinh^-1(x/3)-(sqrt(9+x^2))?

Jul 23, 2018

${\sinh}^{- 1} \left(x\right)$

#### Explanation:

Note that $\left({\sinh}^{- 1} \left(x\right)\right) ' = \frac{1}{\sqrt{{x}^{2} + 1}}$
By the product rule and the chain rule we get

${\sinh}^{- 1} \left(x\right) + x \cdot \frac{1}{\sqrt{{x}^{2} / 9 + 1}} \cdot \frac{1}{3} - \frac{1}{2} \cdot {\left(9 + {x}^{2}\right)}^{- \frac{1}{2}} 2 x$

Simplifying
$x \cdot \frac{1}{\sqrt{{x}^{2} / 9 + 1}} \cdot \left(\frac{1}{3}\right) = \frac{x}{3} \cdot 3 \frac{x}{\sqrt{{x}^{2} + 9}} = \frac{x}{\sqrt{{x}^{2} + 9}}$
we get the result ${\sinh}^{- 1} \left(x\right)$