derivative of inseparable equation

Question

Is there a way to find the derivative of an equation where I cannot separate the independent variable from the dependent variable. For example, what is $dy/dx$ given

\begin{equation*} \cos(y) = \dfrac{x}{x+y} \end{equation*}

Answer 1

Hint: Try implicit differentiation, that is, differentiating the whole thing with respect to $x$ so for example $$\frac{d}{dx}(\cos(y))=-\sin(y) \frac{dy}{dx}$$ Then, after you take the derivative of both sides you can you can solve for $\frac{dy}{dx}$.