Here is a page for explanations of some of the more tricky questions that turn up occasionally.

Why is $\frac{d}{dx}ln(x)=\frac{1}{x}$ ?

To start with, it's worth looking at the graph and seeing that this is reasonable.

So it seems plausible. How about an exact calculation?

We start with $y=ln(x)$ and we want to compute $\frac{dy}{dx}$

Now the really tricky part is that when y is a one-to-one function of x, which it is in this case if we restrict ourselves to positive x, then $\frac{dx}{dy}=1/\frac{dy}{dx}.$ To see that properly you can either use graphs and swap the co-ordinates around, or you can do the limiting process for each side.

Once you accept that, we have

What is 0.999999... actually equal to?

This is a good one. Students seem to think that it can't be one, because it's "obviously" less than one.

How do you convince them otherwise?

What is the value of $0^0$ ??

Hmm ...

Are there any others?

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