Answer:
![\frac{dy}{dx}=\frac{\cos(x)-x\sin(x)}{\sin(y)+y\cos(y)}](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B%5Ccos%28x%29-x%5Csin%28x%29%7D%7B%5Csin%28y%29%2By%5Ccos%28y%29%7D)
Step-by-step explanation:
So we have:
![y\sin(y)=x\cos(x)](https://tex.z-dn.net/?f=y%5Csin%28y%29%3Dx%5Ccos%28x%29)
And we want to find dy/dx.
So, let's take the derivative of both sides with respect to x:
![\frac{d}{dx}[y\sin(y)]=\frac{d}{dx}[x\cos(x)]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5By%5Csin%28y%29%5D%3D%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5Ccos%28x%29%5D)
Let's do each side individually.
Left Side:
We have:
![\frac{d}{dx}[y\sin(y)]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5By%5Csin%28y%29%5D)
We can use the product rule:
![(uv)'=u'v+uv'](https://tex.z-dn.net/?f=%28uv%29%27%3Du%27v%2Buv%27)
So, our derivative is:
![=\frac{d}{dx}[y]\sin(y)+y\frac{d}{dx}[\sin(y)]](https://tex.z-dn.net/?f=%3D%5Cfrac%7Bd%7D%7Bdx%7D%5By%5D%5Csin%28y%29%2By%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Csin%28y%29%5D)
We must implicitly differentiate for y. This gives us:
![=\frac{dy}{dx}\sin(y)+y\frac{d}{dx}[\sin(y)]](https://tex.z-dn.net/?f=%3D%5Cfrac%7Bdy%7D%7Bdx%7D%5Csin%28y%29%2By%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Csin%28y%29%5D)
For the sin(y), we need to use the chain rule:
![u(v(x))'=u'(v(x))\cdot v'(x)](https://tex.z-dn.net/?f=u%28v%28x%29%29%27%3Du%27%28v%28x%29%29%5Ccdot%20v%27%28x%29)
Our u(x) is sin(x) and our v(x) is y. So, u'(x) is cos(x) and v'(x) is dy/dx.
So, our derivative is:
![=\frac{dy}{dx}\sin(y)+y(\cos(y)\cdot\frac{dy}{dx}})](https://tex.z-dn.net/?f=%3D%5Cfrac%7Bdy%7D%7Bdx%7D%5Csin%28y%29%2By%28%5Ccos%28y%29%5Ccdot%5Cfrac%7Bdy%7D%7Bdx%7D%7D%29)
Simplify:
![=\frac{dy}{dx}\sin(y)+y\cos(y)\cdot\frac{dy}{dx}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7Bdy%7D%7Bdx%7D%5Csin%28y%29%2By%5Ccos%28y%29%5Ccdot%5Cfrac%7Bdy%7D%7Bdx%7D%7D)
And we are done for the right.
Right Side:
We have:
![\frac{d}{dx}[x\cos(x)]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5Ccos%28x%29%5D)
This will be significantly easier since it's just x like normal.
Again, let's use the product rule:
![=\frac{d}{dx}[x]\cos(x)+x\frac{d}{dx}[\cos(x)]](https://tex.z-dn.net/?f=%3D%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5D%5Ccos%28x%29%2Bx%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Ccos%28x%29%5D)
Differentiate:
![=\cos(x)-x\sin(x)](https://tex.z-dn.net/?f=%3D%5Ccos%28x%29-x%5Csin%28x%29)
So, our entire equation is:
![=\frac{dy}{dx}\sin(y)+y\cos(y)\cdot\frac{dy}{dx}}=\cos(x)-x\sin(x)](https://tex.z-dn.net/?f=%3D%5Cfrac%7Bdy%7D%7Bdx%7D%5Csin%28y%29%2By%5Ccos%28y%29%5Ccdot%5Cfrac%7Bdy%7D%7Bdx%7D%7D%3D%5Ccos%28x%29-x%5Csin%28x%29)
To find our derivative, we need to solve for dy/dx. So, let's factor out a dy/dx from the left. This yields:
![\frac{dy}{dx}(\sin(y)+y\cos(y))=\cos(x)-x\sin(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%28%5Csin%28y%29%2By%5Ccos%28y%29%29%3D%5Ccos%28x%29-x%5Csin%28x%29)
Finally, divide everything by the expression inside the parentheses to obtain our derivative:
![\frac{dy}{dx}=\frac{\cos(x)-x\sin(x)}{\sin(y)+y\cos(y)}](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B%5Ccos%28x%29-x%5Csin%28x%29%7D%7B%5Csin%28y%29%2By%5Ccos%28y%29%7D)
And we're done!