Q:

What is $dy/dy$?

Say you have the following function:
$$y=x^2+x$$
Then
$$\frac{dy}{dx}=2x+1$$
However, what if you wanted to find $dy/dy$? I differentiated both sides of the original equation with respect to $y$, getting
$$\frac{d}{dy}[y]=\frac{d}{dy}[x^2+x]$$
Now, I’ve always thought that $dy/dy=1$. After all, the derivative of a variable alone with respect to that variable is 1. However, if I apply the differentiation to the RHS, I’ll find that either of the terms are affected by the differentiation. Thus, 
$$\frac{dy}{dy}=x^2+x$$
The RHS here is the original $y$, so
$$\frac{dy}{dy}=y$$
This does not suggest that $dy/dy=1$. Obviously I’m making a mistake somewhere in my reasoning above. Can someone point out where I went wrong? $dy/dy=1$, correct? It does not equal $y$?

A:

You have not applied differentiation to the right hand side in the correct manner. 
When you take the derivative of $y$ with respect to $x$, you must think of $x$ as the independent variable and $y$ as the dependent variable.
On the other hand, when you reverse the roles and take the derivative of $x$ with respect to $y$, then you must think of $y$ as the independent variable and $x$ as the dependent variable. And when you do that then, by applying the chain rule, you get
$$\frac{d}{dy}[x^2 + x] = 2x \frac{dx}{dy} + \frac{dx}{dy}
$$

