How to differentiate $y$ defined by the equation $\sin(x+y) =y^2 \cos x$?

Question

Given: $$\sin(x+y) = y^2 \cos x$$ find $dy/dx$.

$$\cos(x+y)(1+y')= \text{...product rule...}$$

how do we get the left one?

I am looking at the solution. I tried replacing $\sin(x+y)$ with $\sin x \cos y+ \cos x \sin y$

Answer 1

Let $u=x+y$, then $$\frac{\mathrm{d}u}{\mathrm{d}x} = 1 + y'$$ by implicit differentiation. Now apply the chain rule on $\sin u$.