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 $y$?
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$\begingroup$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 $$\frac{d}{dy}[x^2 + x] = 2x \frac{dx}{dy} + \frac{dx}{dy} $$
$\endgroup$ 2 $\begingroup$The mistake is simply in understanding the notation you are using, if $y = x^2 + x$ then $\frac{d}{dy} = \frac{d}{d(x^2 + x)}$, so the LHS is $\frac{dy}{dy} = 1$ and the right hand side is $\frac{d}{dy}(x^2 + x) = \frac{d}{d(x^2 + x)}(x^2 + x) = 1$ so both sides agree.
The point is that in ignoring the relationship between $x$ and $y$ you are not differentiating correctly. Also note that if we really didn't have a link between $x$ and $y$ then $\frac{d}{dy}(x^2 + x)$ would be equal to zero not 1.
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