How can I differentiate x^2+2y=y^2+4 with respect to x?

To differentiate this kind of expression you would need to use implicit differentiation. 

Although it may sound new, you already have all the skills you need to be able to do it. We will differentiate both sides of the expression. 

We will treat the x's as normal. When we encounter terms with y's in them, we will differentiate these terms and multiply each of them by 'dy/dx'. 

So, it will look like this.

Differentiating both sides, we get:

2x+2dy/dx=2ydy/dx

No, to get the derivative, we will simply rearrange the terms, solving for dy/dx:

2dy/dx-2ydy/dx=-2x

(2-2y)*dy/dx=-2x

dy/dx=-2x/(2-2y)

dy/dx=-x/(1-y)

dy/dx=1/(y-1)

Hence, our soultion is dy/dx=1/(y-1).