# How do you differentiate sin(xy)=1/2?

Jan 10, 2017

Here are two ways to do it.

#### Explanation:

The easy (trick) way

From $\sin \left(x y\right) = \frac{1}{2}$ we conclude that $x y = \frac{\pi}{6} + 2 \pi k$ for integer $k$.

So $\frac{d}{\mathrm{dx}} \left(x y\right) = \frac{d}{\mathrm{dx}} \left(\frac{\pi}{6} + 2 \pi k\right)$. That is,

$y + x \frac{\mathrm{dy}}{\mathrm{dx}} = 0$, and

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

The non-easy way

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

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

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

Divide by $\cos \left(x y\right)$ (OK, since we know it is not $0$.)

to get

$y + x \frac{\mathrm{dy}}{\mathrm{dx}} = 0$, and

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