# Common Maths Questions

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.

• When x is very small, ln(x) is very negative and growing quickly
• so the derivative is large positive.
• When x=1, ln(x) is zero and growing gently.
• so the derivative is about 1, although not necessarily exactly so.
• As x gets large, ln(x) grows more slowly, but always grows
• so the derivative is positive, but getting close to 0.
So it seems plausible. How about an exact calculation?

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

• $y=ln(x)$
• => $x=e^y$
• => $\frac{d}{dy}(x)=\frac{d}{dy}(e^y)$
• => $\frac{dx}{dy}=e^y$ because $\frac{d}{dy}e^y=e^y$
• => $\frac{dx}{dy}=x$ because $x=e^y$
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

• $\frac{dy}{dx}=1/x$
• Hence $\frac{d}{dx}ln(x)=1/x$

## 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?

Hmm ...

CategoryMaths