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Alright, so we haven't even covered this material in class yet, but I remember from a short lesson in pre-calculus that this is a form of the difference quotient.

The problem asks: let $f(x) = \ln(x^3+1)$ and $a = 2$. Use numerical approximation to find the value of: $\lim_{h\to0} {f(a+h)-f(a)\over (h)}$

I would greatly appreciate any tips, since I honestly do not know how to use both an '$a$' value and an $f(x)$ value in this. Lastly, I'm trying to go through the formatting to make sure everything is posted right, so apologies if the formatting is wonky when you read it. Thanks to anyone who replies.

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    Haha, it's nice to see that you at least attempted to do the MathJax in some parts. :-) – 2017-01-22
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    Always have to give some effort! Thank you for helping correct the format Simply Beautiful Art. – 2017-01-22
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    I guess $x=a=2$ & you are supposed to substitute smaller & smaller values into the limit definition of the derivative ... observe that it is tending to a limit & hence $f^{'}(2)=?$ ! ... Yeah thank you SBA for the edit – 2017-01-22

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Well, plugging it all in, we get

$$\lim_{h\to0}\frac{\ln((2+h)^3+1)-\ln(2^3+1)}h$$

Plotting a few points:

$$\begin{array}{c|c}h&\Delta f\\\hline1&1.134979933\\0.1&1.311257235\\0.01&1.33111237\\?&?\\-0.01&1.335556765\\-0.1&1.355652055\\-1&1.504077397\end{array}$$

Taking the average of the two middle points, I get

$$f'(2)\approx1.333334568$$

In comparison to the actual value:

$$f'(2)=\frac43\approx1.3333333333\dots$$

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    and all done with a handy dandy scientific calculator :-) – 2017-01-22
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    Thank you for your help, I really appreciate it! – 2017-01-22
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    @GottaLoveArchery No problem. :-) – 2017-01-22