Finding Instantaneous Velocity: Taking the Derivative?

[Femme_physics]

Taking the derivative seems to be the easiest way of dealing with solving for instantaneous velocity. Should I ever bother with taking the reciprocal, simplifying, or factoring the expression in order to find the 0/0 culprit and cancel it out, or should I just take the derivative each and every time?

If so, why do we even bother learn these other long methods of finding the derivative where we can just use the derivative relatively simple rules to find it, and then we can find easily find the limit as x goes to 0, thereby finding our instantaneous velocity.

 

[chiro]

Taking the derivative seems to be the easiest way of dealing with solving for instantaneous velocity. Should I ever bother with taking the reciprocal, simplifying, or factoring the expression in order to find the 0/0 culprit and cancel it out, or should I just take the derivative each and every time?

If so, why do we even bother learn these other long methods of finding the derivative where we can just use the derivative relatively simple rules to find it, and then we can find easily find the limit as x goes to 0, thereby finding our instantaneous velocity.

Calculus is a generic tool and finding instantaneous velocity from a displacement graph is just one application of the differential calculus.

One thing that you should notice is that with limits is that the limit term (usually h) approaches zero but is never zero. 0/0 is not defined at all but some small number e/e is always 1 even when e approaches zero (but it can never ever take on the value 0).

I agree with your frustration about spending months (maybe years in some cases) of learning the "bag of tricks" that is tricks and transforms to solve calculus problems: personally I find it kinda pointless sometimes trying to find some obscure trick to solve some stupid DE, but the truth is that math heavy scientists often need to do this.

I guess an analogy is like learning arithmetic instead of being introduced a calculator. We don't often go as far as some people go (like some chinese kids that have to do intense training with abacus systems), but I would worry if any kid didn't do some kind of thorough introduction to arithmetic the "old-fashioned" way that kids in primary school have to do.

Just remember with your limit, you can only cancel out your h^n/h terms since they will always end up in unity. If you want to know the reason, one reason is due to l'hospital's rule for limits.

Also remember what the concept of differentiation is: if you realize that, then any other application of calculus should help you realize why you have to use the "rules" that you are taught.

 

[Femme_physics]

Thanks for the comprehensive reply. I think I rather just take the derivative by using the power/prouct/chain/quotient rules forever. I'm not a mathmatician, I'd like to understand the stuff I need and no more so I could get on with physics and my mechatronics course, but if I spend too much time simplifying complicated terms where I can just use those 4 basic derivative rules I might needlessly overcomplicate my studies. Just my philosophy. Again, thanks for the reply.

I'd like to add that my current position can change, but if the analogy is genuinely equivalent to that of a calculator usage for arithmetic, then I'll stick to the easier method.

 

[jtbell]

Taking the derivative seems to be the easiest way of dealing with solving for instantaneous velocity. Should I ever bother with taking the reciprocal, simplifying, or factoring the expression in order to find the 0/0 culprit and cancel it out, or should I just take the derivative each and every time?

Your first method sounds like setting up $\Delta x / \Delta t$ explicitly, and taking the limit as $\Delta t \rightarrow 0$. The only reason you're doing that is as an example of applying the general definition of the derivative, so you can verify that the "cookbook rules" for powers, products, etc. actually work as advertised. After you've done it once or twice, there's no reason to do it again.

 

[Femme_physics]

I see. Thanks a bunch jtbell :)

 

[General_Sax]

After you've done it once or twice, there's no reason to do it again.

Are you saying that I've been wasting my time all these years!?

 

[CJSGrailKnigh]

Ya you really have been... Maple can even do series solutions so why bother once you get your degree. Even in 3rd year my profs accept my math proofs from MATLAB and maple runtimes. But that may not apply outside of Eng... I don't know. Thats not to say it isn't necessary to understand what and how maple does it