If $dx/dy =\sin(x)$ then is $dy/dx = 1/\sin(x)$?

If $\dfrac{dx}{dy} = \sin(x),$ then is $\dfrac{dy}{dx} = \dfrac{1}{\sin(x)}$?

I'm trying to understand how to manipulate $dx$ and $dy$ quantities effectively.

• If Panny walks ten times the speed of Nanny, then does Nanny walk one-tenth of the speed of Panny? Dec 14, 2014 at 17:29
• @Venus Could you please provide that link for everyone's benefit? An obvious place to expect an admonishment about "too much vertical space" would be in meta.math.stackexchange.com/questions/9959/… where I see a request to minimize vertical space in titles, but nothing like that for the question itself. Dec 15, 2014 at 14:19
• Stop defacing this question. I'm removing unrelated comments. Dec 15, 2014 at 15:07

By the chain rule (assuming all quantities exist and make sense):

$$\dfrac {\mathrm dx}{\mathrm dy}\dfrac {\mathrm dy}{\mathrm dx} = \dfrac {\mathrm dx}{\mathrm dx} = 1$$

edit: the OP asked when this works. This works if $y$ is invertible and the derivative isn't $0$. If $y$ is not invertible, then $x$ might still implicitly define a differentiable function of $y$ for some neighborhood of a point you're interested in.
• It works. :) But can I always do that? For any given $y = f(x)$? Dec 15, 2014 at 16:54