Double differential / differential squared

**wglmb** · February 3rd 2010, 07:46 AM

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

**HallsofIvy** · February 3rd 2010, 08:11 AM

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

**wglmb** · February 3rd 2010, 08:20 AM

Thanks a lot

**Danny** · February 3rd 2010, 03:51 PM

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 $<br /> \frac{\displaystyle \frac{d}{dt}\left(\displaystyle \displaystyle \frac{dr}{dt}\bigg /\displaystyle \frac{d \theta}{dt}\right)}{\displaystyle \frac{d \theta}{dt}}<br />$ .

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