How do you derive differential area without specifying a variable?

AI Thread Summary
The discussion revolves around the confusion regarding the use of derivatives and differentials in mathematical expressions. Participants clarify that the notation used in textbooks often omits explicit variables, leading to misunderstandings about whether derivatives are being taken with respect to something specific. The distinction between differentials and derivatives is emphasized, with the suggestion that terms can cancel out when all are expressed with respect to the same variable. An example involving the product rule and time derivatives is provided to illustrate the concept. Overall, the conversation highlights the importance of understanding the context and rules governing the use of differentials in calculus.
mfc5200
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Can someone please explain the steps to derive this? I've seen it everywhere, but the explanation is never given.

My confusion arises from the second line. In EVERY math book I've ever picked up, whenever you take a derivative, it is ALWAYS with respective to something, i.e y w.r.t x, or some variable w.r.t time, etc. But in this example as shown in textbooks, it seems as though they are taking a derivative, but it is not w.r.t anything. For example, instead of saying dy/dx, they would just be saying dy, which doesn't really make any sense to me. I'm confused about this.

I haven't seen a good explanation of this type of derivative yet. I attached an example. Thanks
 

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It's just a differential area. You have not paid close enough attention to your math books. :wink:
 
Last edited:
Look up the product rule for derivatives. The dot implies the time derivative. Since all terms on both sides are wrt dt it cancels out.
 
FredGarvin said:
Since all terms on both sides are wrt dt it cancels out.

Could you maybe say a little bit more about that. I think this is the part that is confusing me.
 
It's a differential, it's not a derivative.
 
mfc5200 said:
… But in this example as shown in textbooks, it seems as though they are taking a derivative, but it is not w.r.t anything. For example, instead of saying dy/dx, they would just be saying dy, which doesn't really make any sense to me. I'm confused about this.
Cyrus said:
It's a differential, it's not a derivative.

Hi mfc5200! :wink:

Does this help … http://en.wikipedia.org/wiki/Differential_(infinitesimal) ? :smile:
 
Suppose

m-dot =pVA
then
d(m-dot)/dt = pV dA/dt + VA dp/dt + pA dV/dt
Now multiply both sides by dt.
 
Bob S said:
Suppose

m-dot =pVA
then
d(m-dot)/dt = pV dA/dt + VA dp/dt + pA dV/dt
Now multiply both sides by dt.

Yea, I was able to get that far. I just wasn't sure if you were "allowed" to do that. I've never seen that done before.
 
mfc5200 said:
Yea, I was able to get that far. I just wasn't sure if you were "allowed" to do that. I've never seen that done before.

You're not always allowed to do this. So, be aware of that.
 

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