# Given that  sin(x/y) = 1/2  find dy/dx?

Nov 1, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y}{x}$

#### Explanation:

We have:

$\sin \left(\frac{x}{y}\right) = \frac{1}{2}$

Differentiating Implicitly wrt $x$ and applying the chain rule and the product rule we get:

$\setminus \setminus \setminus \setminus \cos \left(\frac{x}{y}\right) \left\{\frac{d}{\mathrm{dx}} \left(\frac{x}{y}\right)\right\} = 0$

$\therefore \cos \left(\frac{x}{y}\right) \frac{\left(y\right) \left(\frac{d}{\mathrm{dx}} x\right) - \left(\frac{d}{\mathrm{dx}} y\right) \left(x\right)}{y} ^ 2 = 0$

$\therefore \cos \left(\frac{x}{y}\right) \frac{\left(y\right) \left(1\right) - \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) \left(x\right)}{y} ^ 2 = 0$

$\therefore \cos \left(\frac{x}{y}\right) \frac{y - x \frac{\mathrm{dy}}{\mathrm{dx}}}{y} ^ 2 = 0$

$\therefore y - x \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\therefore x \frac{\mathrm{dy}}{\mathrm{dx}} = y$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y}{x}$