# What is the derivative of (arcsin(3x))/x?

Apr 23, 2015

You can solve this problem by using the quotient rule .

$y = \frac{\arcsin \left(3 x\right)}{x} = \frac{u}{v}$

$u = \arcsin \left(3 x\right)$

$\sin u = 3 x$

$\cos u \cdot \frac{\mathrm{du}}{\mathrm{dx}} = 3$

$\frac{\mathrm{du}}{\mathrm{dx}} = \frac{3}{\cos} u = \frac{3}{\sqrt{1 - 9 {x}^{2}}}$

$v = x$, $\frac{\mathrm{dv}}{\mathrm{dx}} = 1$

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

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