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?
What is the value of $0^0$ ??
Hmm ...
Are there any others?
CategoryMaths
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