# Functions and derivatives using limits

For a calculus project, I need to create a function and find its derivative using the limit definition of derivatives.

I could use a function from this list, but I want to know how I could come up with my own function, $f$, where the derivative of $f$ can be found algebraically using the limit definition of derivatives:

Edit: After simplification and factoring, I'd like to be able to get a factor of $\Delta x$ out of the top $f(x + \Delta x) - f(x)$ so that I can remove the denominator.

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This is not exactly a question. It's actually the suggestion to the problem you have. Be sure that you ask a question instead of getting suggestions from people. Asking this way makes the site less of Q&A forum. – NasuSama Sep 30 '13 at 1:16
No, I'm not asking to be provided with a function. I'm wondering how I could generate a function that meets the criteria. – Joel Christophel Sep 30 '13 at 1:35
Regarding your recent edit, that is true. If the function is differentiable, that will happen ($\Delta x$ will vanish in the denominator by way of a factor in the numerator). It is all about skill, and some differentiable functions are harder than others. – J. W. Perry Sep 30 '13 at 2:56
I see. Thank you. I tried to find the derivative of the simple quadratic $8x^2 + 9x$ but got stuck after I got to $24x^2 + 8\Delta x^2 + 18x + 17\Delta x$ – Joel Christophel Sep 30 '13 at 3:02
I would show that by doing $8x^2$, then $9x$, as the limit of the sum equals the sum of the limits. You can do it your way though and I like it your way too, just remember you are shooting for $16x+9$, so having the target in mind always helps :). – J. W. Perry Sep 30 '13 at 3:04

This is a purposefully incomplete answer (don't want to spoil your project), and some bit of an explanation. A function $f$ is differentiable if the derivative exists at every point in its domain (a seemingly pitiful definition but true). If a function is differentiable, then your limit definition $$f'(x)=\lim_{\Delta x \rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ holds true. Your ability to actually show this for the harder cases however will depend on your skill level, and so I expect that trying to prove the easier, as well trying to approach a few of the harder differentiable functions, will be a part of your project.

A stick in the spoke so to speak: All differentiable functions are continuous, but not all continuous functions are differentiable. For a function $f$ to be differentiable at every point in its domian there is more burden. This should be a part of your project (if it is not then just show the derivative of $f(x)=x$ and sit back down). You should attempt to explain that extra burden. You have probably already seen links like wikipedia, and your textbook. This is really the greatest part of your project; when is a function differentiable? If you can explain that in a way that your peers can understand, then you have an awesome project! Just read the literature and do your best.

As it turns out, all polynomials are differentiable. Show an example of a linear function, then a quadratic (more work, but not beyond you). Then try to approach $f(x)=\sin(x)$, then try some logarithms like NasuSama said.

I would personally stick to "parent like" monomial functions like $x$, $x^2$, $\sin x$, $\ln x$, $e^x$..., as the limit of the sum equals the sum of the limits, and that is not hard to show (you can make that another project). Keep us posted. I would really like to see you post a self answer to this after you finish your project. I would vote that up big time!

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This is not an answer; it's actually the suggestion.

There are many examples, such that the derivative is found by the limit definition of derivative. Here is one example from the list you provided:

$$f(x) = \cos(3x)$$

Few good examples to go for are $\ln(\ln|x|)$ and $7\ln|\sec(x) + \tan(x)|$. It's all up to you to find the better example rather than the ones I have for you.

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