Definition of $\frac{dy}{dx}$ where we cannot write $y=f(x)$ Informally, I can say that "for a small unit change in $x$, $\frac{dy}{dx}$ is the corresponding small change in $y$". This is however a bit vague and imprecise, which is why we have a formal definition, such as: 
Where $y=f(x)$, we define $$\frac{dy}{dx}=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}. $$
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But what about if we are unable to express $y$ as a function of $x$? (To pick a random example, say we have $\sin (x+y) + e^y = 0$.)
Then what would the general definition of $\frac{dy}{dx}$ be? (How would the above definition be modified?)
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To elaborate, applying the $\frac{d}{dx}$ operator to my random example, we get $$\left(1+\frac{dy}{dx}\right)\cos(x+y)+e^{y}\frac{dy}{dx}=0 \implies \frac{dy}{dx}=\frac{-\cos(x+y)}{\cos(x+y)+e^{y}}.$$
Again, I can say that "for a small unit change in $x$, $\frac{dy}{dx}$ is the corresponding small change in $y$". But what would the corresponding precise, formal definition be?

There does exist a $y(x)$ such that $\sin{(x+y(x))} = e^{y(x)}$ is true for all $x$ in a (sensibly chosen) open interval $I$. [1] We may not be able to write it down, but it exists. We can do calculus on this $y$, for which the derivative $y'(x) = \lim_{a\rightarrow x} \frac{y(a) - y(x)}{a-x}$ is well-defined, but (again) not (immediately) expressible in terms of elementary functions. We can, in this case, find an expression for $y'(x)$ by means of the generalised chain rule, because $R(x,y) = \sin{(x+y)} + e^y$ is expressed in terms of elementary functions and its partial derivatives are easy to find.
[1] Actually, there exist an infinity of such $y$, since $\sin{(x+y+n\pi)} = \sin{(x+y)}$ and so we choose a $y$ corresponding to a particular value of $n$.
