# Thread: Double differential / differential squared

1. ## Double differential / differential squared

If $r$ and $\theta$ are funtions of $t$, does $\frac{d^2 r}{dt^2}\bigg / \left ( \frac{d\theta}{dt}\right ) ^2$ equal $\frac{d^2 r}{d\theta ^2}$?

2. Originally Posted by wglmb
If $r$ and $\theta$ are funtions of $t$, does $\frac{d^2 r}{dt^2}\bigg / \left ( \frac{d\theta}{dt}\right ) ^2$ equal $\frac{d^2 r}{d\theta ^2}$?
No. While the first derivative can be "treated like" a fraction, the second derivative cannot.

For example, take $r= t^2$ and $\theta= t^4$. Then $\frac{dr}{dt}= 2t$, $\frac{d\theta}{dt}= 4t^3$, $\frac{d^2r}{dt^2}= 2$ and $\frac{d^2\theta}{dt^2}= 12t^2$.

Now it is easy to see that $\theta= (t^2)^2= r^2$ so $\frac{d\theta}{dr}= 2r= 2t^2= \frac{4t^3}{2t}$ $= \frac{\frac{d\theta}{dt}}{\frac{dr}{dt}}$.

But $\frac{d^2\theta}{dr^2}= 2$ while $\frac{\frac{d^2\theta}{dt^2}}{\frac{d^2r}{dt^2}}= \frac{12t^2}{2}= 6t^2$.

3. Thanks a lot

4. Originally Posted by wglmb
If $r$ and $\theta$ are funtions of $t$, does $\frac{d^2 r}{dt^2}\bigg / \left ( \frac{d\theta}{dt}\right ) ^2$ equal $\frac{d^2 r}{d\theta ^2}$?
To find $\frac{d^2r}{d \theta^2}$ you would need to calculate $
\frac{\displaystyle \frac{d}{dt}\left(\displaystyle \displaystyle \frac{dr}{dt}\bigg /\displaystyle \frac{d \theta}{dt}\right)}{\displaystyle \frac{d \theta}{dt}}
$
.