1

If $x=y^2$, then $\frac{dx}{dy}=2y$, which means that the rate of change of $x$ with respect to $y$ equals $2y$.

We can also find the differential of $x$. We get that $dx=2ydy$. But what is this with respect to? It doesn't make any sense to have a rate of change without respect to anything, right?

James Ronald
  • 2,331
  • $dx$ is an infinitesimal change in $x$, not a rate – J. W. Tanner May 13 '20 at 18:41
  • I prefer to think of $dx$ and $dy$ as one-forms on $\mathbb{R}^2$ - take a look at some introductory differential geometry for a more complete picture. e.g. Loring Tu's book Intro to Manifolds – DanLewis3264 May 13 '20 at 18:44

2 Answers2

1

It means that if $y$ changes by a small amount $dy$, then the corresponding change in $x$ will be $2ydy$. One can divide by $dt$, where $t$ is an arbitrary variable and change the equation to $$\frac{dx}{dt} = 2y \frac{dy}{dt}$$ This shows that the change in $x$ will be the same, no matter what we're differentiating with respect to.

Vishu
  • 14,469
0

The statements $dx=2ydy$ and $\frac {dx}{dy}=2y$ are the same. The first one is the rate of change of $x$ with respect to $y$ so if you change your y from $y_0$ to $y_0+dy$ then the linear approximation to change in $x$ is $dx=2ydy$

Mohammad Riazi-Kermani
  • 68,728