# How do you find the derivative of 3arccos(x/2) ?

Aug 4, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{3}{\sqrt{4 - {x}^{2}}}$

#### Explanation:

$y = 3 {\cos}^{-} 1 \left(\frac{x}{2}\right)$

$x = 2 \cos \left(\frac{y}{3}\right)$

Differentiate x with respect to y

$\frac{\mathrm{dx}}{\mathrm{dy}} = - 2 \sin \left(\frac{y}{3}\right) . \left(\frac{1}{3}\right)$

$\frac{\mathrm{dx}}{\mathrm{dy}} = - \left(\frac{2}{3}\right) \sin \left(\frac{y}{3}\right)$

We Need to find $\frac{\mathrm{dy}}{\mathrm{dx}}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{3}{2 \sin \left(\frac{y}{3}\right)}$

$\frac{y}{3} = {\cos}^{-} 1 \left(\frac{x}{2}\right)$
dy/dx=-3/(2sin(cos^-1(x/2))
dy/dx=-3/(2sin (sin^-1((sqrt(4-x^2))/2))
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{3}{\sqrt{4 - {x}^{2}}}$