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

##### 1 Answer
Jul 31, 2014

First recall the definition for derivative of $\arcsin x$:

$\frac{d}{\mathrm{dx}} \left[\arcsin x\right] = \frac{1}{\sqrt{1 - {x}^{2}}}$.

Since we're differentiating with $\frac{x}{3}$ instead of $x$, we need to substitute and apply the chain rule:

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

Simplifying yields:

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

A page explaining this simplification in more detail can be found here .