OK a spot of A-level maths comes to the blog

A few things to remember:

- Ln x – The Natural log of x – is the inverse function
^{Footnote 1}of e^{x} - d/dx e
^{x }= e^{x }– That’s what so special about e^{x } - Remember the result from last time that

Now how can we use these three facts to find d/dx ln x?

Let f(x) = y = Ln x . We want f’(x) = dy/dx ^{Footnote 2}

Raise e to both sides

e^{y }= e^{ln x } = x (Using 1 above)

so x = e^{y} so dx/dy = e^{y }(Using 2 above)

so dy/dx = 1/ e^{y}

e^{y} = x so substitute this in the give f’(x) in terms of x and we get 1/x.

**d/dx ln x = 1/x.**

Footnotes

1 I use the word function here but caution is required. You must be careful with the domain of Ln x if it’s going to meet the criteria of a Function – It must be defined as x≥0

2. Ok so I’m mixing my calculus formats here! Both have their uses, maybe the subject for another blog entry.