# Thread: xsin(y) = cos(x+y) , then find y'

1. ## xsin(y) = cos(x+y) , then find y'

Ive been banging my head on this one for 30 minutes..

xsin(y) = cos(x+y) , then find y'

2. Originally Posted by alny
Ive been banging my head on this one for 30 minutes..

xsin(y) = cos(x+y) , then find y'
$\frac{d}{dx} (x \sin y) = \frac{d}{dx} \cos(x+y)$

First, the left side using the product rule:

$\frac{d}{dx} (x \sin y) = \frac{d}{dx} (x) \cdot \sin y + x \cdot \frac{d}{dx} (\sin y) = \sin y + x \cos y \, \frac{dy}{dx}$

Now the other side:

$\frac{d}{dx} \cos(x+y) = -\sin(x+y) \cdot \frac{d}{dx} (x+y) = -\sin(x+y) \cdot \left(1+\frac{dy}{dx}\right)$

So we now have:

$\sin y + x \cos y \, \frac{dy}{dx} = -\sin(x+y) \cdot \left(1+\frac{dy}{dx}\right)$

At this point is it an algebra problem to solve for $\frac{dy}{dx}$. Can you finish this?