Derivative of a reciprocal function If I want to prove that a partial derivative $\frac{\partial D}{\partial x}$ is either positive or negative, and $D=\frac{1}{E}$, then how can I use the partial derivative $\frac{\partial E}{\partial x}$ (which is a simplier derivative) to prove this?
Since $\;\cfrac{\partial D}{\partial x}=\cfrac{\partial}{\partial x}\left(\cfrac{1}{E}\right)=-\cfrac{1}{E^2} \cfrac{\partial E}{\partial x}\;,\;$ then
$\;\cfrac{\partial D}{\partial x}&gt;0\;\iff\;\cfrac{\partial E}{\partial x}&lt;0\;,$
$\;\cfrac{\partial D}{\partial x}&lt;0\;\iff\;\cfrac{\partial E}{\partial x}&gt;0\;.$
So, if you want to prove that $\;\cfrac{\partial D}{\partial x}\;$ is positive, it is sufficient to prove that $\;\cfrac{\partial E}{\partial x}\;$ is negative, and vice versa.
I don't think that is possible because continuity of the function itself might change.
For example
If your function is $xy$, then upon taking the reciprocal, $1/xy$ isn't continuous at $x=0$ and $y=0$. So ideally being able to reciprocate to differentiate will not give you an answer.
Also, let us assume the domain isn't an issue for us. Still, assume a function y=x which is a straight line with a positive slope. If you rewrite it as $y=1/x$, it is a  rectangular hyperbola that has a negative slope. So, this as per my knowledge doesn't seem possible