If you use the limit method for calculating a derivative, the denominator is the difference between x and (a) as (a) approaches x. But that limit would be equal to zero. So how is it that you can actually find the derivative since you would have to divide by zero?
Did that make sense?
I did too. Sadly, not matter what method I use, I tend to go back to wanting to use L'Hopital's rule (which requires me to already know the derivative of ). lol.
I don't know if this derivative was ever actually shown to me using the definition of the derivative.
CrazyAsian, I don't think you're missing anything obvious becuase if so, Jhevon and I are missing that same thing.
If then .
Now the is called by Serge Lang the mysterious limit. There is a good but informal discussion of that fact in Lang’s A First Course in Calculus. This is by no means easy to prove. In beginning courses, it is discussed informally.
i think i got something. it may be a bit choppy though.
Recall that where is a function of
Note that:
Now consider
Now apply the definition:
The reason i do not like this though is that i think the relationship only works if is a linear function of
EDIT: Nevermind, is a linear function! since is a constant. So what do you guys think?
As long as is a constant, img.top {vertical-align:15%;} \cdot e^u" alt="\frac {d}{dx} e^u = \frac {d}{dx} \cdot e^u" /> is true. u does not have to be a linear function.
The problem I had with doing this is that we would have to show using the definition of the derivative is true.
good to know
yea, but we have to draw the line somewhereThe problem I had with doing this is that we would have to show using the definition of the derivative is true.
i just thought that would be a part of the background knowledge of someone wanting to prove ...even though i didn't know that fact as well as i should. so i kind of treated that fact as a proven lemma that didn't need to be proven again. i guess we could prove it before though...do i know how to prove that?
Using the definition is the bad way to do it.
For example,
Try doing that with the defintion!!
Instead, we use more powerful techiniques which are proven by the definitions.
is by definition . We can find the derivative of this using the chain rule since is differenciable as is differenciable everywhere. So .