# What is the derivative of sin^-1 * (5x)?

May 3, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{5}{\sqrt{1 - 25 {x}^{2}}}$

#### Explanation:

$y = {\sin}^{- 1} \left(5 x\right)$

$\implies 5 x = \sin y$

differentiate$\text{ " wrt" } y$

$5 \frac{\mathrm{dx}}{\mathrm{dy}} = \cos y$

$\frac{\mathrm{dx}}{\mathrm{dy}} = \cos \frac{y}{5}$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{5}{\cos} y$

we need to rewrite this in terms of the original variable $x$

now:

${\sin}^{2} y + {\cos}^{2} y = 1$

$\implies \cos y = \sqrt{1 - {\sin}^{2} y}$

$\implies \cos y = \sqrt{1 - {\left(5 x\right)}^{2}}$

$\therefore \cos y = \sqrt{1 - 25 {x}^{2}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{5}{\sqrt{1 - 25 {x}^{2}}}$